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DUKE 

UNIVERSITY 

LIBRARY 

Treasure  %oom 

1' 


THE 


FEDERAL   ACCOUNTANT: 

CONTAINING, 

I.  COMMON  ARITHMETIC,  the  Rules  and  Illustrations. 

II.  EXAMPLES  aod  ANSWERS  with  BLANK  SPACES,  sufficient  for 
their  operation  by  the  Scholar. 

III.  To  each  Rule  a  SUPPLEMENT,  comprehending,  1.  QUESTIONS 

on  the  oMature  of  the  Rule,  its  Usk  and  the  nianuer  of  its  Operations. 
2.  EXERCISES. 

IV.  FEDERAL  MONEY,  with  rules  for  all  the  various  operations  in  it. 

to  reduce  Federal  to  Oi.b  Lawful,  and  Old  Lawful  to 
FEDERAL  MONEY. 

V.  Interest  cast  in  Federal  Mo.vey,  with  Compound  Multiplication , 
Compound  />irmo7(,and  Practice, -wrought  in  Old  LAwrur  and  in  Federal 
MoNEv  ;  the  same  questions  being  put  in  separate  columns  on  the 
same  page  in  rach  kind  of  money,  these  two  modes  of 
account  become  contrasted,  and  the  great  advan- 
tage gained  by  reckoning  in  Federal 
Money  easily  discerned. 

VI.  Demonstrations  by  Engravings  of  the  reason  and  nature  of  the 

various  stops  in  the  eKtraction  of  the  Square  and  Cube  Roots,  not  to 

b»  found  in  any  other  treatise  on  Arithmetic. 

Vn.  Forms  of  Notes,  Deeds,  Bokds  and  other  Instruments  of  Writirg. 

THE  WHOLE  IN  A  FORM  AND  METHOD  ALTOGETHER*  NEW,  FOR  THE 

EASE  OF  THE  MASTER  AND  THE  GREATER  PROGRESS 

OF  THE  SCHOLAR. 


BY  DANIEL  ADAMS,  M.  B. 


STEREOTYPE  EDITION, 

KETISED    AND   CORRECTED,   WITH   ADDITIONS. 


KEENE,  N.  H— PRINTED  BY  JOHN  PRENTISS, 

[proprietor   of    THE    COPY    RIGHT.] 

Sold  at  his  Bookstore,  and  by  the  principal  Booksellers  in  the  New-England  Stat«»- 
'  and  New- York. —  IS)9. 
— «=>*-o— 

'Price  10  Dollars  per  liostn,  1  Dollar  siuglt. 


NcivhaDrpshirc  Dialricty  ss. 


I^li  IT  RKMEMBERED,  that  on  the  seventeenth  day  of  July,  in  tLe  thirty 
L.  s.  niuth  year  of  the  Imlcpendence  of  the  United  States  of  America,  Daniel 

Adams,  of  Mont  Vernon,  in  said  District,  hath  deposited  in  tiiis  office  the  Title  of 
a  Book,  tlje  riijht  whereof  Ire  claims  as  Author,  in  the  following  words,  to  wit : — "  The 
Scholar's  .ifitfttnelic. :  or  Federal  Accountant,  (.'ontaining  I.  Common  Arithmetic,  the 
Rules  and  Illustrations. — II.  Examples  and  Answers,  with  blank  spaces  sufficient  for  their 
operation  by  Ori  Scholar. — III.  To  each  Rule,  a  Supplement,  comprehending,  1.  Questions 
on  (lie  nature  of  the  rule,  its  use,  and  the  manner  of  its  operations. — 2.  Exercises. — IV. 
Federal  Monej',  with  rules  for  all  the  various  operations  in  it,  to  reduce  Federal  to  Old 
Lawful,  and  Old  Lawful  to  Federal  Money. — V.  Interest  cast  in  Federal  Money  with 
("om|»ouiid  Multiplication,  Compound  Division  and  Practice  wrought  in  Old  Lawful  and 
ill  Federal  Money,  the  same  questions  being  put  in  separate  columns  on  the  same  page  in 
each  kind  of  money,  by  which  these  two  modes  of  account  become  contrasted,  and  the 
great  advantage  gained  by  reckoning  in  Federal  Money  easily  discerned. — VI.  Demon- 
strations by  engravings  of"^  the  reason  and  nature  of  the  various  steps  in  (he  extraction  of 
the  S(juai"e  and  Cube  Roots,  not  to  be  found  in  any  other  treatise  on  Arithmetic. — VII. 
Forms  of  Notes,  Deeds,  Bonds,  and  other  instruments  of  writingr — The  whole  in  a  form 
and  method  altogether  new,  for  the  ease  of  the  Master  and  the  greater  i)rogi'es3  of  the 
Scholar.— Bv  DANIEL  ADAMS,  M.  B." 

In  conformity  to  the  act  of  Congress  of  the  United  States,  entitled  '•  an  act  for  the 
encouragement  of  Learning,  by  securing  the  copies  of  Maps,  Charts  and  Books,  to  the 
authors  and  proprietors  therein  mentioned,  and  extending  the  benefit  thereof  to  the  ails 
of  Dfsiguiug,  Engraving,  Etching,  Historical  and  other  prints. 

G.  AV.  PRESCOTT,  Clerk  of  tht  U.  S.  Court,  JV.  H.  Disfrict. 


PREFACE. 


IT  13  fourteen  years  siace  the  first  Edition  of  the  Scholar's  Arithmetic 
was  offered  to  the  Public.  It  has  now  gone  through  nine  editions,  and 
more  than  Forty  Tliousand  copies  have  been  circulated.  In  those  places 
where  it  has  been  introduced,  it  never  has,  to  the  best  of  our  knowledge, 
been  superseded  by  any  other  work  which  has  come  in  competition  with  it. 
A  knowledge  of  these  facts  is,  perhaps,  one  of  the  best  recommendations 
which  can  be  desired  of  the  work. 

It  has  now  undergone  a  careful  revisal.  Some  of  the  rules  have  been 
thought  to  be  deficient  iii  examples ;  in  tliis  revised  edition,  more  than  six- 
ty new  examples  have  been  added  under  the  different  rules.  Some  have 
expressed  a  desire  that  answers  might  be  given  to  the  "  Miscellaneous 
Questions,'"  at  the  end  of  the  book  ;  these  have  been  added  accordingly, 
and  the  number  of  these  questions  increased.  But  what  more  particularly 
claims  attention  in  this  revised  edition,  is  the  introduction  of  the  rule  of 
Exchange,  where  the  pupil  is  made  acquainted  with  the  different  curren- 
cies of  the  several  states,  (that  of  S.  Carolina  and  Georgia,  only  excepted,) 
and  how  to  change  these  currencies  from  one  to  another  ;  also,  to  Federal 
Money,  and  Federal  Money  to  these  several  currencies.  This  has  been 
done  more  particularly  with  a  view  to  the  accommodation  of  the  State  of 
New-York,  and  other  more  southern  states,  where  this  work  has  already 
acquired  a  very  considerable  circulation.  Answers  are  given  to  many  of 
the  questions  in  different  currencies,  so  that  the  pupil  in  N.  England, 
N.  York,  &:c.  will  find  an  answer  to  the  question,  each  in  tlie  currency  of 
his  own  particular  state. 

These  comprehend  the  only  additions  in  the  present  new  edition. 

Wo  have  now  the  testimony  of  many  respectable  Teachers  to  believe;, 
that  this  work,  where  it  has  been  introduced  into  Schools,  has  proved  a 
vpry  kind  assistant  towards  a  more   speedy  and  thorough  improvement  of 


IT  PKEFACE. 

Scholars  in  Numbers,  and  at  the  same  time,  has  relieved  masters  of  a  hea?y 
burden  of  writing  out  Rules  and  Questions,  under  which  they  have  so  long 
labored,  to  the  manifest  neglect  of  other  parts  of  their  Schools. 

To  answer  the  several  intentions  of  this  work,  it  will  be  necessary  that 
it  should  be  put  into  the  hands  of  every  Arithmetician  :  the  blank  after 
each  example  is  designed  for  the  operation  by  the  scholar,  which  being 
(irst  wrought  upon  a  slate,  or  waste  paper,  he  may  afterwards  transcribe 
into  his  book. 

The  Supplements  to  the  Rules  in  this  work  are  something  new  ;  ex- 
perience has  shown  them  to  be  very  useful,  particularly  those  "  Questions,^' 
unanswered,  at  the  beginning  of  each  Supplement.  These  questions  the 
pupil  should  be  made  to  study  and  reflect  upon,  till  he  can  of  himself  devise  , 
the  proper  answer.  They  should  be  put  to  him  not  only  once,  but  again, 
and  again,  till  the  answers  shall  become  as  familiar  with  him  as  the  num- 
bers in  his  multiplication  Table.  The  Exercises  in  each  supplement  may 
be  omitted  the  first  time  going  through  the  book,  if  thought  proper,  and 
taken  up  afterwards  as  a  kind  of  review. 

Tlirough  the  whole  it  has  been  my  greatest  care  to  make  myself  intelli- 
gible to  the  scholar  ;  such  rules  and  remarks  as  have  been  compiled  from 
other  authors  are  included  in  quotations  ;  the  Examples,  many  of  them  are 
extracted  ;  this  I  have  not  hesitated  to  do,  when  I  found  them  suited  to  my 
purpose. 

Demonstrations  of  the  reason  and  nature  of  the  operations  in  the  ex- 
traction of  the  Square  and  Cube  Roots  have  never  been  attempted  in  any 
work  of  the  kind  before  to  my  knowledge  ;  it  is  a  pleasure  to  find  these 
have  proved  so  higldy  satisfactory. 

Grateful  for  the  patronage  this  work  has  already  received,  it  remains 
only  to  be  observed,  that  no  pains  nor  exertions  shall  be  spared  to  merit  iti 
continuance. 

DANIEL  ADAMS. 

MoiU-Vcrnnu,  (A'.  //.)  December  26th,  1815. 


RECOMMENDATIONS. 


JVew-Salm,  Sept.  14f/«>  180!. 
HAVING  attentively  examined  "  The  Scholar's  Ariihmf.Hc,"  I  cheerfully  give  it  as  my 
opinion  tliat  it  is  well  calculated  for  the  instruction  of  youth,  and  that  it  will  abridge  muoh 
of  the  time  now  necessary  to  be  spent  in  the  communication  and  attainment  of  such 
Arithmetical  kno^vledse  as'is  proper  for  the  discbarge  of  business. 

WARREN  FIERCE. 
Preceptor  of  A'eic-5a/eni  Academy- 


Crohn  Jicadtmy,  Sept.  2,  1801. 

Sir I  liave  pnniscd  with  attention  "  The  Scholar's  Arithmetic,"  which  you  transmitted 

to  me  some  time  since.  It  is  in  my  opinion,  better  calculated  to  lead  students  in  our 
bchools  !uid  Academies  into  a  complete  knowledge  of  all  that  is  useful  in  that  branch  of 
lilcratun  ,  ilinn  any  other  work  of  the  kind  I  have  seen.  With  great  sincerity  I  wi.sh  you 
siiccess  in  your  exertions  for  the  promotion  of  useful  learning  ;  and  I  am  confident  that  to 
be  generally  approved  your  work  needs  only  to  be  generally  known. 

WILLIAM  M.  RICHARDSON, 
Preceptor  of  the  Academy. 


Extract  of  a  Letter  from  the  Hon.  JoHi»  Wheelock,  Lt.  D.  President  of  Dartmouth  College, 

to  the  Author. 
"  The  Scholar's  Arithmetic  is  an  improvement  on  former  productions  of  the  same  nature. 
Its  distinctive  order  and  ?upplement  will  help  the  learner  in  his  progress;  the  part  on  Fe- 
deral Money  makes  it  more  useful ;  and  I  have  no  doubt  but  the  whole  will  be  a  new  fund 
of  profit  in  our  country." 


September  "Jfh,  1807. 

The  Scholar's  Aritiimcllc  contains  most  of  fne  important  Rules  of  the  Art,  and  some- 
thing, al?o.  of  (he  curious  and  entertaining  kind. 

Tlie  subjects  are  handled  in  a  simple  and  concise  manner. 

While  the  questions  arc  few,  they  exiiibit  a  ccnsideral.-le  varietj'.    While  they  are  gene- 
rally easy,  .some  of  them  afford  .=cope  for  tlic  exercise  of  ihc  Pniiohir's  judgnicnt 

It  is  a  good  (juality  of  the  Book,  that  ithas.=:o  much  to  do  with  Federal  Money. 

The  plan  of  sliow  in^  the  reasons  of  the  ojwjrations  in  the  extraction  of  (i-.e  Squard  and 
Cube  Roots  is  good.  DAiMEL  HARDY,  .lus. 

Preceptor  of  Chesterfield  Academy. 


Exlracl  oj  a  LeHer  fiom  thz  Rev.  Laban  Ainswohtti  cf  -Jtiffrcy,  to  (he  publish^  of  (he 
fuurlh  Edition,  dated  August  3,  1S07. 
'•  The  snperioriiy  of  the  Scholar's  Arillimetic  to  any  book  of  (!k^  1  ind  in  ray  knowledge, 
( learly  appears  from  its  good  rlfect  in  fhe  schools  I  aimually  visit. — Previous  lo  its  intm- 
<luction,  AritliHielic  v/as  learned  and  performed  mcclinrHcaliy  ;  since,  scholars  are  able  to 
z'\'  >"  r»  i..!ioniil  r.ccoisnt  of  the  seveial  c|)('r;itions  in  Arithia'j'iir;  whi'oli  is  the  fjst  proof  of 
"i^ir  having  jparned  togoed  purj'osn  ' 


CONTENTS. 


Inlroduclioa P^gf  7 

>'otation 7 


Numeration Poge  8 

Explanation  of  Characters    .....  10 


SECTION  I. 

Fundamental  Rules  of  .'irithmelic. 


Simple  Addition 11 

do.      Siitdrarlion     .......    16 

do.     MultipliciUion 19 


Simple  Division     .     . 
Compound  Addition 
do.        Subtraction 


27 
35 
46 


SECTION  II. 


Pules  tssenlially  necessary  for  every  person  to  fit  and  qualify  than  for  the  transaetion  of 

business. 
Rfduction 51  I  Method  of  casfinsr  Interest  on 


Fractions 68 

DerimnI  Fractions 69 

Fedf-ral  Money 8f) 

Kxcliarige 84 

Tal)lf  (o  reduce  .shiilinj^s  and  ^ 
pence  to  cents  and  mills       ^    * 

Tables  of  Exchange 92 

Interfst 9.3 

Easy  mctiiod  of  casting  Interest  ...  97 


91 


99 


Notes  and  Bonds  when  par-  f 
tial  payments  at  diflerent  / 
times  have  been  made         J 

Compound  Interest 104 

do.        Multiplication       ....  H)5 

do.        Division 110 

Single  fiule  of  Three HT 

Double  Rule  of  Three l.W 

Practice       HI 


SECTION  HI. 

Rales  occasionally  vstfai  lo  men  in  parliculur  rmplouncnts  of  life. 


Involution 15" 

Evolution 1.57 

Extraction  of  the  Square  Root  .     .     .    1§8 
Demonstration  of  the  reason  and  \ 
nature  of  the  various  steps  in 
the  operation  of  extracting  the 
S()UHre  Root 
Extraction  of  the  Cube  Root 
Demonstration  of  the  reason  and 
nature  of  the  various  steps   in 
the  operation  of  extracting  the 
Cube  Root  1 

Sinjile  Fellowship 177 

Double  Fellowship 171> 

Barter 182 

Loss  and  Gain 185 

Duodecimals 188 


159 


167 


168 


Examples  for  measuring  wood   .     .     .  189 

Boards      .     .190 

Painter's  and  Joiner's  work    ....  192 

Glazier's  work 192 

Alligation 193 

Medial 193 

Alternate 194 

Position       198 

Single 198 

Double 199 

Discount 201 

Equation  of  paymeHls 201 

Guaging       203 

Mechanical  Powers       203 

The  Lever        203 

The  Axle 204. 

The  Screw 204 

2M 


PEr.BLEMs  1st.  To  find  (he  circumference  of  a  circle,  the  diameter  being  given    - 

2d.  To  find  the  area  of  a  circle,  (he  diameter  being  given 264 

3d.  To  measure  the  solidity  of  an  irregular  body '•-  204 

Bliscellaneous  Questions  --..----, 206 

SECTfON  IF. 
Form  of  Koles.  ^c. 


Notes 211 

Bonds 212 

Ficceipls        2)."-{ 

Orders 211 


Deeds      -     -     -    ., 214 

Indenture      .--.-----     215 
Will 216 


THE 

SCHOLAR'S  ARITHMETIC 

INTRODUCTION. 


ARITHMETIC  is  the  art  or  science  which  treats  of  numbejF?. 

It  is  of  two  Iciods,  theoretical  and  practical. 

The  Theory  of  Arithmetic  explains  the  nature  and  quality  of  number*?, 
and  demonstrates  the  reason  of  practical  operations.  Considered  in  this 
sense,  Arithmetic  is  a  Science. 

Practical  Arithmetic  shews  the  method  of  working  by  numbers,  so 
as  to  be  most  useful  and  expeditious  for  business.  In  this  sense  Ajrithmetic 
is  an  A7-t. 

DIRECTIONS  TO  THE  SCHOLAR. 

Deeply  impress  your  mind  with  a  sense  of  the  importance  of  arithmetical 
knowledge.  The  great  concerns  of  life  can  in  no  way  be  conducted  without 
it.  Do  not,  therefore,  think  any  pains  too  great  to  be  bestowed  for  so  noble 
an  end.  Drive  far  from  you  idleness  and  sloth  ;  they  are  great  enemies  to 
improvement.  Remember  that  youth,  like  the  morning,  will  soon  be  past, 
and  that  opportunities  once  neglected,  can  never  be  regained.  First  of  all 
things,  there  must  be  implanted  in  your  mind  a  fixed  delight  in  study  ; 
make  it  your  inclination  ;  "  A  desire  accomplished  is  s-xeet  to  the  soul.''''  Be 
not  in  a  hurry  to  get  through  your  book  too  soon.  Much  instruction  may  be 
given  in  these  few  words,  UNDERSTAND  f.vf.ry  tiiin(;  as  you  go  Ai.oNt;. — 
Each  rule  is  lirst  to  be  committed  to  memory  ;  afterwards,  the  examples  in 
illustration,  and  every  remark  is  to  be  perused  with  care.  There  is  not  a 
word  inserted  in  this  Treatise,  but  with  a  design  that  it  should  be  studied 
by  the  Scholar.  As  much  as  possible,  endeavour  to  do  every  thing  of  your- 
self;  one  thing  found  out  by  your  own  thought  and  rcllection,  will  be  of 
more  real  use  to  you,  tiian  'tzccnti/  (Iiings  told  yon  by  an  Instructor.  Be  not 
overcome  b}'  little  seeming  dilTicullies,  but  rather  strive  to  overcome  such 
by  patience  and  application  j  so  shall  3'our  progress  be  easy  and  the  object 
of  your  endeavours  sure. 


Off  entering  upon  this  most  useful  study,  the  first  thing  which  the  Scholar 
lias  to  regard,  is 

NOTATION. 

Notation'  is  the  art  of  expressing  numbers  by  certain  characters  or  fig- 
ures : '  of  which  there  are  two  methods.;  1.  The  Unman  mclltiMt'hy  Letters. 
-.   The  .Arabic  riietlioif,  by  i'igi-res.     The  hitter  ii'  that  of  ge^fifl  u-.e. 


8  INTROpUCTIOX. 

In  the  Arabic  method  all  numbers  are  expressed  by  these  ten  characters 
or  figures. 

1  2  .3  4567890 

Unit;  or     two  ;  three  ;  four  ;    five  ;     six  ;  seven  ;  eight  j  nine  ;  cypher 
o"<^    .  [or  nothing-. 

The  nine  first  are  called  si^niJicaiU,  figures,  or  digits,  each  of  which 
standing  by  itself  or  alone,  invariably  expresses  a  particular  or  certain  num- 
ber ;  thus,  1  signilies  one,  2  signijjes  t-xo,  3  signifies  three,  and  so  of  the 
rest,  until  you  come  to  nine,  but  for  any  number  more  than  nin£,  it  will 
always  require  two  or  more  of  those  figures  set  together  in  order  to  ex- 
press that  number.     This  will  be  more  particularly  taught  by 

NUMERATION. 

•  Nameration  teaches  how  to  read  or  tc-r/fc  any  sum  or  number  by  fio-ures. 

In  setting  down  numbers  for  arithmetical  operations,  especially  with  be- 
ginners, it  is  usual  to  begin  at  the  ri^U  hand,  and  proceed  towards  the  left. 

Example.  If  you  wish  to  write  the  sum  or  number  537,  begin  by  setlini^ 
down  the  seven,  or  right  hand  figure,  thus  7,  next  set  down  (he  three,  at  the 
left  band  of  the  sevon,  thus  37,  and  lastly  the  fie,  at  the  left  hand  o(  the 
three,  thus  537,  which  is  the  number  projiosed  to  be  written. 

In  this  sum  thus  written  you  are  next  to  obsei"ve  that  there  are  three  places, 
meaning  the  situations  of  the  three  different  figures,  and  that  each  of  these 
places  has  an  appropriated  name.  The  frst  place,  or  that  of  the  right  hand 
'  figure,  or  the  place  of  the  7,  is  called  vnit^s  place;  the  accond  place,  or  that 
of  the  figure  standing  next  to  the  right  hand  figure,  in  this  the  jdace  of  the  3, 
is  called  ten's  place  ;  the  third  place,  or  next  towards  the  left  hand,  or  place 
of  the  5,  is  called  hundred's  place ;  the  next  or  fourth  place,  for  we  may 
suppose  more  figures  to  be  connected,  is  thousand's  place ;  the  next  to  this 
tens  of  thousand's  place,  and  so  on  to  what  length  we  please,  there  being 
particular  names  for  each  place.  Now  every  ti^ure  signifies  difTerentlv,  ac- 
cordingly as  it  may  happen  to  occupy  one  or  the  other  of  these  places. 

The  value  of  the  first  or  right  hand  figure,  or  of  the  figure  standing  in  the 
place  o(  units,  in  any  sum  or  number,  is  just  what  the  figure  expresses  stand- 
ing alone  or  by  itself;  but  every  other  figure  in  the  sum  or  number,  or  those 
to  the  left  hand  of  the  first  figure,  have  a  dilferent  signification  from  their 
true  or  natural  meaning ;  for  the  next  figure  from  the  right  baud  towards 
the  left,  or  that  figure  in  the  place  of  tens,  expresses  so  many  times  ten,  as 
the  same  figure  signifies  units  or  ones  when  standing  alone,  that  is,  it  is  te?i 
times  its  simple  primitive  value  ;  and  so  on,  every  removal  from  the  right 
hand  figure,  making  the  figure  thus  removed  ten  times  the  value  of  the  same 
figure  when  standing  in  the  place  immediately  preceding  it. 


Example.  Take  the  sum  3  3  3,  made  by  the  same  figure  three  times 
repeated.  The  first  or  right  hand  figure,  or  the  figure  in  the  place  oi'  units, 
has  its  natural  meaning  or  the  same  meaning  as  if  standing  alone,  ar.d  signi- 
fies three  units  or  ones  ;  but  the  same  figure  again  towards  the  left  hand  in 
the  second  place,  or  place  of  tens,  signifies  not  three  units,  but  three  tens,  that 
is  thirty,  its  valne  being  increased  in  a  tenfold  propoi-tion ;  proceeding  on  still 
further  towards  the  left  hand,  the  next  figure  or  that  in  the  third  place,  or 
place  onnmdreds  signifies  neither  three  nor  thirty,  but  three  hundred,  which 
is  ten  times  the  value  of  that  figure,  in  the  jdace  immediately  preceding  it,  or 
that  in  the  place  oiLiens.     So  you  might  proceed  and  add  the  figarc  3,  iifty  or 


INTRODUCTION.  9 

an  hundred  times,  and  every  time  the  figure  nas  added,  it  would  signify 
ten  times  more  than  it  did  the  last  time. 

A  Cypher  standing  alone  is  no  signitication,  yet  placed  at  the  right 
Iland  of  another  figure  it  increases  the  value  of  that  figure  in  the  same  ten- 
fold proportion,  as  if  it  had  been  preceded  by  any  other  figure.  Thus  3, 
sfanding  slone,  signifies  three ;  place  a  cypher  before  (30)  and  it  no  longer 
signifies  three,  but  thirty ;  and  another  cypher  (300)'%nd  it  signifies  three 
lionrlrefl. 

The  value  of  figures  in  conjunction,  and  how  to  read  any  sum  or  num- 
ber agreeably  to  the  foregoing  observations,  may  be  fully  understood  by  the 
following 

TABLE. 
ri  -:  ^  The  words  at  the  head  of  the  Table  shew 

■^  ^  i  ^  rt  ^^^  signification  of  the  figures  against  which 

'^  C  ;2  .2  g  ^  they  stand  ;  and  the  figures  shew  how  many 

,•,    •  S -^  2      -a  §  of  that  signification  are  meant.     'Thns  Units 

§=£^^2      Ha  in  the  first  place  sio-nify  ones,  and  6  standinjf 

£"^°^i3       <^-^OT    .  against  it,  shews  that  six  ones  or  individuals 

.t«^'c-5!£    •-p^'c-S  arc  here   meant ;  tens  in  the  second  place 

c'^oS-iJoc-iofj;?:;        .  shew  that  every  fitjure  in  this  place  means  so 

P  -a    '«    3  -TD     -fi  .2  "3    tfi     S  '^     -/•   jn  _  J    o      .        J-  ■        ^     -J.  I 

uacaocK^acocG.t:  many  tens,  and  3  stanamfr  aaramst  it,   shews 
cq^U(t_i2jf_,;52^f_U(^E-(ti  *hat  three  tens  are  here  meant,  equal  to  thir- 
21G7235  4  21836  iy,what  the  figure  really  signifies.  Hundreds 
3407G214G312in  the  third  pl«ce  shew  the  meaning  of  fig- 
13025037645  ures  in   this   place   to  be  Hundreds,  and  8 
41393210G4  shews   that  eight  hundreds  are  meant.     In 
27021367  5  the  same  manner  the  value  of  each  of  the  re- 
46327291  maining  figures  in  the  table  is  known.    Har- 
12  3  4  6  3  2  ing  proceeded  through  in  this  way,  the  sum 
2  3  4  5  6  7  of  the  first  line  of  ligures  or  those  immedi- 
8  9  0  9  8  ately  against  the  words,  will  be  found  to  be 
7  6   5  4  Tv:o  Billions,   one  hundred  sixty  seven  thou- 
12  3  sands,  izi'o  hundred  and  tluriy-Jisc  Millions  : 
4  5  four  Jiundred  tzt:c7ity-one  thousands  ;  eight  hun- 
7  dred  and  thirty-six.  In  the  like  manner  may  be 
rerii]  all  the  remaining  numbers  in  the  Table. 
Those  words  at  the  head  of  the  Tnble  are  applicable  to  any  sum  or  num- 
ber, and  must  be  committed  perfectly  to  memory  so  as  to  be  readily  appUed 
on  any  occasion. 

For  the  greater  case  of  reckoning,  it  is  convenient  and  often  practised  in 
public  ofiices,  and  by  men  of  business,  to  divide  any  number  into  periods 
and  half  periods,  as  in  the  following  manner  : 

5.  3  7  9,  G  3  4.  5  2   1,  7  6  8.  5  3  2,  4  6  7 


fiO        C^        ^       Co 


■^  "^  "^  ^ 


C;=:;=3=;i?        S£:5-C-CJ        eaa*:Hi2 

^  -G  -C)  ~Ci  --i       .     o^S^  e     r*     £  3    ^    S     S 

-c;   5   e  ^-  c-i   -^    c    -   ?:    .^  ;^   ~  T3   s: 


„    <o  ^  <; 


;-  C^     -  I  g  S^  ^ 


lu  INTRODUCTION. 

The  first  six  fisrxires  (v<m\  the  tijrht  hand  are  railed  the  unit  period,  thf 
next  six  the  viillion  peri<M,  aller  wliich  the  lrillio)i,  quadrillion,  quini'uliot 
periods.  &.c.  follow  in  th<Sr  ordeiC 

Thus  by  the  use  of  ten  figures  may  be  reckoned  every  thing  which  cti'< 
be  numbered  ;  things,  the  multitude  of  which  far  exceeds  the  oouaprehcu 
nion  of  man. 

'*  It  may  not  be  aihiss'ln  illustrate  by  a  few  examples  the  extent  of  nu'P- 
"  bors,  which  are  frequently  named  without  beinf:^  attended  to.  if  a  pii 
"  son  employed  in  telling  money,  reckon  an  hundred  pieces  in  a  minute, 
"  and  continue  at  work  ten  hour^  each  da}',  he  will  take  seventeen  days 
*'  to  reckon  a  million  ;  a  thousand  men  would  take  45  years  to  reckon  .1  bil- 
"  lion.  If  we  suppose  the  whole  earth  to  be  as  well  peopled  as  Britain, 
"  and  to  have  been  so  from  the  creation,  and  that  the  whole  race  of  mankind 
"  hnd  constantly  spent  their  time  in  teiline;  from  a  heap  consisting  'f  a 
*•  (juadriliion  of  pieces,  they  would  hardly  have  yet  reckoned  a  thousanijth 
••  part  of  that  quantity." 

Afier  having  been  al»le  to  read  correctly  to  his  instructor,  all  the  nnm- 
i>crs  in  the  I'oregoinj^  Table,  the  learner  may  proceed  to  write  the  follow- 
ing nnmbers  out  in  figures. 

Two  hundred  and  sixty-three. 

^  Five  thousand  one  hundred  and  sixty. 

One  hundred  thousand,  six  hundred  and  four. 

Five  miHion,  eighteen  thousand,  seven  hundred  &  six. 

Two  million,    six  hundred  and  fifty  thousand,  one 

hundred  and  thirty-seven. 
Seven  hundred  and   ninety-four  million,  one  hun- 
dred   and    forty-nine  thousand,    live    hundred 
and  twenty. 
Three   thousand,  nine   hundred  and  foity  million, 
four  hundred  and  two  thousand,  eight  hundred 
and  four. 
Five  hundred  thirty  six  thousand,  two  hundred  and 
seventy  two  million,  one   hundred  and  three 
thousand  and  six. 
^  ^  Four  })ill'.on,  six  hundred  thousand  million,  seven 

<  hundred  thousand,   two  hundred  and  ninety- 

4  f  two. 

^*  ■  — 

Explanation  of  the  CJiaracters  made  use  of  in  this  Work. 

__  ^  The  sign  of  equality  ;  as  100  cts=l  Dol.  signifies  that  100  centg^ 
f  are  equal  to  1  dollar.  , 

^      Saint  Gporge's  Cross,  the  sign  of  addition  ;  as  2-|- 1=6,  that  is,  "i 
"^      I  addpd  to  4  ;)r-j  equal  to  6.  .  J 

—  The  sign  ofi^-ubt.artion  ;  as 6 — 2=4;  thatis,2  takenfrom  61eareK4. 
^  Saint  Aniircw's  Cross,  the  sign  of  multiplication  ;  as  4X  G=~i  ; 
(  that  is,  4  times  6  r:re  equ?l  to  24. 

w       i      Reveiscd  rarentheees,  the  sign  of  divismn  ;  as  3)6(2,  that 
^^      I  is,  6  divided  by  3  the  qiKti3nt  is  2,  or  G-i-  .3=2.      - 

The  s^on  of  proportioi.  ;  as  2  :  4  t :  8  •  16,  that  is,  as  2 
to  4  so  is  0  to  lti»  r 


X 


SECTION  I. 


FUNDAMENTAL  RULES  OF  ARITHMETIC. 

1  HESE  are  four,  Additiox,  Subtraction,  Multiplica- 
tion, and  DIVISIO^';  they  may  be  either  simple  or  compotmdj 
simple,  when  the  numbers  are  all  of  one  sort  or  denomina- 
tion ;  compound,  when  the  numbers  are  of  different  deno- 
minations. 

TuEY  are  called.  Principal  or  Fundamental  RuleSy  be- 
cause all  other  rules  ancl  operations  in  arithmetic  are 
not!iin2;  more  than  various  uses  and  repetitions  of  these 
four  rules. 


Thfi  object  of  every  aritlnnetiral  operation,  is,  hy  certain  given  qnantilips  which  arr 
known,  to  find  otit  others  which  ;ire  iinknowti.  This  cannot  be  done  hut  by  changes  ef 
fectcd  on  the  given  niinibcrs  ;  and  as  the  only  way  in  whicii  numbers  can  be  changed  i'- 
either  by  increasins;  or  diminishing^  tlieir  (juantitie?,  and  as  there  can  be  no  increase  or 
diminution  of  numbers  but  by  one  or  the  other  of  the  al)ove  operations,  it  consequently 
lollows,  tiiat  tlusc  full)-  rules  embrace  the  whole  art  of  .\rilhmctic. 


^  I.  SIMPLE  ADDITION. 

SiMFi.E  Addition  is  the  putting  together  of  two  or  more  number?,  of  the 
same  denoiflhiation,  so  as  to  make  them  one  whole  or  total  number,  Called 
the  sran,  or  amount. 

RULE. 

ilPl^ce  the  numbers  to  be  added  one  under  another,  with  units  under 
unit?,  lens  tinder  tens.  &.c.  and  draw  n  line  under  the  lowest  number. 

2.  Add  the  right  hand  column,  and  if  the  sum  be  under  ten,  write  it  under 
the  col'imn  ;  but  if  it  be  ten,  or  any  exact  number  of  tens,  rxrite  a  cypher  ; 
and  if  it  he  not  sn  exact  number  of  tens,  write  the  excess  above  tens  at  the 
foot  of  (lie"  rolnmn,  and  for  every  tin  the  sum  contstns,  carry  one  to  the  next 
column,  and  add  it  in  the  same  manner  as  the  former. 

3.  Proceed  in  like  manner  to  add  the  other  columns,  carrying  for  the 
tej^iS  of  each  to  the  next,  and  set  down  the  full  sum  of  the  last  or  left  hand 
column. 

PROOF. 

Reckon  the  figures  from  the  top  downwards,  and  if  the  work  be  right, 
this  amount  will  be  equal  to  the  first ; — or,  what  is  often  practised,  "  cut 
"  off  the  upper  line  of  figures  and  (ind  the  amount  of  the  rest ;  then  if  the 
*'  amount  and  upper  line  when  adrjed,  be  equal  to  the  stim  total,  the  work 
"  is  supposed  to  be  right.' 


12 


SIMPLE  ADDITION. 
EXAMPLES. 


Sect.  I.  1. 


1.  What  will  be  ^  |  E^  il 
the  amount  of  ...  3  6   1   2  dolls, 
dollars  Avhen  added  together  ? 


»  s   oJ  -2 
2  5  ?  "S 

8  0  4  3 


dolls. 

o 


6  5  1  dolls,  and  of  3 


tS 


3 


a 
6 


1    2   dollars 


8 

0 
6 

4 
6 

3 

1 
3 

dollars 
dollars 
dollars 

1     2 

3 

0 

9 

dollars 

8 

6 

9 

7 

Here  are  four  sums  given  for  addition  ; 
two  of  them  contain  V7iits,  tens,  hundreds,  "©^ 
tJioiisands  ;  another  of  them  contains  units,  ^ 
lens,  hundreds;  and  a  fourth  contains  units 
onh'.  The  first  step  to  prepare  these  sums 
for  the  ojieration  of  addition,  is  to  write  them 
(loun,  units  under  units,  tens  under  tens,  ^c. 
thus  ; — 

Answer,  or  Amount, 

Amount  of  the  three  lower  lines, 

Proof,  12     3    0    9 

To  find  the  answer  or  amount  of  the  sums  given  to  be  added,  begin  with 
the  right  hand  column,  and  say,  3  and  1  make  4,  and  3  are  7,  and  2  are  b  ; 
xvhich  £um  (9)  being  less  than  ten,  set  down  directly  under  the  column  you 
added.  Then  proceeding  to  the  next  column,  say  again;  5  and  4  are  9, 
and  1  is  10  ;  beinj;  even  ten,  set  down  0,  and  carry  one  to  the  next  column, 
saying  1,  v/hich  I  carry  to  6  is  7,  and  0  is  nothing,  but  6  make  13  ;  which 
sum  (13)  is  an  excess  of  3  over  even  ten  ;  therefore  set  down  3  and  carry 
1  for  the  ten  to  G  in  the  next  column,  saying  1  to  8  is  9,  and  3  are  1'2  ;  this 
being-  the  last  cohimn,  set  down  the  whole  number  (12)  placing  the  2,  or 
unit  figure  directly  under  the  column,  and  carrj'ing  the  otiier  figure,  or 
the  1,  forward  to  the  next  place  on  the  left  hand,  or  to  that  of  Tens  of 
thousands,  and  the  work  is  done. 

It  may  now  be  required  to  know  if  the  woiv.  be  right.  To  exhibit  the 
method  of  proof  let  the  upper  line  of  figures  be  cut  off  as  «een  in  the  ex- 
.'imple.  Then  adding  the  three  lower  lines  which  remain,  place  the  amount 
(.jG97)  imdeF  the  amount  first  obtained  by  the  addition  of  all  the  sum^-',  ob- 
serving carefully  that  each  figure  fail  directly  under  the  column  which  pro- 
iluced  it;  then  add  this  last  amount  to  the  upper  line  which  you  cut  ofi'; 
llius  7  to  2  are  9  ;  9  to  1  are  10  ;  carry  one  to  6  is  7  and  6  are  13  ;  2„jv!vi*'.h 
1  carry  again  to  8  is  9  and  3  are  12,  all  vv'hirh  being  set  down  in  their  pw'pcv 
t'liices,  and  as  seen  in  the  example,  compare  the  amount  (12309)  Jast  o!)- 
itined,  wiih  the  first  amount  (12.309)  and  if  they  agree,  as  it  i^i  seen  in  thif« 
case  they  do,  then  the  work  is  judged  to  be  right. 

Note. — The  reason  of  carnjing  for  ten  in  all  simple  numbers  is  evident 
from  v.'hat  has  been  taught  in  Notation.  It  is  because  10  in  an  inf-'iioj 
.  oiiunn  is  just  equal  in  value  to  1  in  a  superior  column.  As  if  a  man  shov.ld 
te  holding  in  his  ri^ht  liand  half  pistareen-,  and  in  his  left,  dollais.  !l  you 
should  take  10  half  pistareens  ivum  Iris  right  lir.nd,  and  put  one  dollar.into 
his  left  hand,  you  would  not  rob  the  nsan  (.f  any  of  his  money,  because  1  ot 
those  pieces  in  his  left  hand  is  just  equal  iu  vulue  to  10  of  tiiosa  in  h;^ 
y\'>A\'.  hand. 


Sect.  I.  1. 

SIMPLE  ADDITION. 

13 

2 

3 

4 

= 

= 

=3 

1     6     7 

5 

2 

6 

0 

3 

7 

3 

4 

7     1 

o 

G 

5     4     0 

3 

8 

2 

4 

8 

0 

5 

7 

0     3 

2 

2     6     3 

7 

1 

2 

6 

5 

1 

4 

2 

1     G 

8 

3 

1 

6 

4 

3 

7 

0 

1 

3 

6 

2 

1 

5 

2 

4 

3 

6 

3 

5 

4 

2 

8 

6 

6 

3 

0 

9 

8 

1 

7 

5 

2 

1 

3 

2 

4 

3 

2 

2 

1 

4 

3 

1 

0 

6 

'i 

7 

5 

2 

6 

3 

1 

9 

8 

5 

1 

7 

6 

4 

5 

0 

6 

3 

9 

8 

7 

5 

1 

3 

4 

5 

6 

7 

8 

9 

2 

4 

6 

8 

2 

3 

7 

6 

9 

8 

6 

5 

3 

5 

1 

0 

2 

8 

7 

5 

4 

1 

0 

7 

3 

8 

7 

9 

T) 

2 

8 

6 

7 

8 

5 

4 

0 

3 

9 

6 

7 

4 

9 

8 

0 

1 

2 

6 

4 

7 

3 

1 

0. 

] 

3 

2 

0 

1 

3 

3 

6 

9  10 

4261783  653865076 

6402531  983124 

3  52643  40750208 


7 

6 

2 

3 

4 

1 

6 

3 

4 

1 

2 

5 

0 

7 

8 

0 

1 

7 

6 

0 

Cf 

4 

6 

8 

2 

2 

9 

2 

0 

1 

1 

3 

5 

3 

6 

4 

o 
O 

3 

5 

6 

8 

7 

3 

6 

1 

0 

7 

0 
3 

__ 

14  SUPPLEMENT  TO  ADDITION  Sect.  I.  1. 

SUPrLEMENT  TO  ADDITION. 


THK  altPnlive  t-cholar  nho  has  understood,  and  still  carries  in  his  mind, 
what  Iras  alrea«ly  been  tanj^ht  him  of  Addition,  will  be  able  to  answer  his 
instructor  to  the  following 

QUESTIONS. 

L  What  is  simple  Addition  ?  ^ 

a.  How  do  you  place  numbers  to  be  added  ? 

3.  Where  do  you  be^in  the  addition  ? 

4.  What  is  the  answer  called  ? 

5.  How  is  the  sum  or  amount  of  each  column  to  be  set  down  ? 

6.  AVhat  do  you  observe  in  regard  to  setting  down  the  sum  of  the  last 

column  ? 

7.  Why  do  you  carry  for  ten  rather  than  any  other  number  ? 

8.  How  is  addition  jproved  ? 

NoTi:.  Should  the  learner  find  any  difficulty  in  giving  an  answer  to  the 
above  questions,  he  is  advised  to  turn  back  and  consult  his  Rule,  with  its 

illustrations. 

EXERCISES. 

1.     What  i^  the  amount  of  2801  2.    Suppose  you  lend  a  neighbour 

dollars  ;    765  dollars  ;    and    of  397  £210  at  one  time,  £,1G  at  another, 

rlo]':.:>,  when  added  together  ?  £17  at  another,  and  £,9  at  another. 

Ans.  3963  dollars.  What  is  fhe  sum  lent  ?  Ms.  £312. 


Note.  The  scholar  who  looks  at  greatness  in  his  class,  will  not  be  dis- 
couraged by  a  little  difficulty  which  may  at  first  occur  in  stating  his  ques- 
Uoii,  but  will  apply  himself  tlic  more  closely  to  his  Rule,  and  to  thinking. 
:hat  if  pos-^ible  he  may  be  able  himself  to  aiiswer  what  another  may  be 
obliged  to  have  taught  him  by  his  instructor. 

o.  A  tree  was  broken  off  by  the  4.  A  ma«  being  asked  his  age, 
\Aw\,  27  feet  from  the  ground  ;  the  said  he  was  27  years  old  when  he 
port  broken  off  was  71  feet  long  ;  married,  and  he  had  been  married 
vhat  was  the  hei«rht  of  that  tree  be-  J5  years.  What  was  the  man's 
fore  it  wa=  broken  ?  ag«  ? 

.'3«f   OS  frr,^ 


Sect.  I.  1. 


SUPPLEMENT  TO  ADDITION. 


15 


5.  Washington  was  born  in  the  6.  There  are  two  numbers  ;  tkc 
year  of  our  Lord  1732  ;  he  was  67  less  number  is  3761,  the  difference 
years  old  when  he  died  ;  in  what  between  the  numbers  is  597  ;  what 
year  of  our  Lord  did  he  die  ?  ^is  the  greatest  number  ? 

Ans.  1109.  'w  Ans,  9352. 


7.  From  the  Creation  to  the  de- 
parture of  the  Israelites  from  Egypt 
was  2513  years  ;  to  the  siege  of 
Troy,  307  years  more  ;  to  the  build- 
ing of  Solomon's  temple,  180  years  ; 
to  the  building  of  Rome,  251  years  ; 
to  the  expulsion  of  the  kings  from 
Rome,  244  years  ;  to  the  destruc- 
tion of  Carthage,  363  years  ;  to  the 
death  of  Juhus  Cajsar,  102  years  ; 
to  the  Christian  aera,  44  years ;  re- 
quired the  time  from  the  Creation  to 
the  Christian  sera  ? 

Am.  4004. 


8.  At  the  late  Census,  talien 
A.  D.  1810,  the  number  of  inhabit- 
ants in  the  several  jXew-England 
States  was  as  f^illows  ;  viz.  Maine, 
228705;  N.  Hampshire,  214460; 
Vermont,  217895  ;  MassaGhusctti, 
472040  ;  Rhode-bland,  76931  ;  Coti- 
?iecticut,  261942;  what  was  the  num- 
ber of  inhabitants  at  that  time  in 
New-England  ? 

Ans^  1471973. 


9.  Thefc  are  five  numbers,  the 
first  is  2617;  the  second  893;  the 
third  1702 ;  the  fourth  as  much  as  the 
three  first ;  and  the  fifth  twice  as 
much  as  the  third  and  fourth  ;  what 
is  the  wh©Ie  sum  ? 

An^.  24252. 


10.  A  gentleman  left  his  son, 
2475  dollars  more  than  his  daughters, 
whose  fortune  was  25  thousand,  25 
hundred,  and  25  dollars  ;  what  was 
the  son's  portion,  and  what  was  the 
amount  of  the  whole  estate  ? 

Ans.  Son's  portion,  30000. 
Whole  estate.  57525. 


IQ  SIMPLE  SUBTRACTION.  Sect.  I.  % 


i  2.  SIMPLE  SUBTRACTION. 


SIMPLE  SUBTRACTION  is  taking  a  less  number  from  a  greater  of 
the  same  denomination,  so  as  to  shew  the  difierence  or  remainder  ;  as  3 
taken  from  8,  there  remniins  3.  ' 

The  greater  number  (8)  is  called  the  Minuend,  the  less  number  (5)  the 
S:illrah4:n(1,  and  t)>e  difference  (3)  or  what  is  left  after  Subtraction,  the 
I'ernuitider. 

RULE. 

"  Place  the  less  number  under  the  greater,  units  under  units,  ten?  under 
tens,'  and  so  on.  Draw  a  line  bc'ow  ;  then  bcg-in  at  the  right  hdm",  and 
gul»lract"each  iijjure  of  the  less  number  from  the  iigure  above  it,  and  piace 
ti:e  remainder  directly  below.  When  the  tigure  in  the  lawer  line  exceeds 
th^  figure  ybove  it,  suppose  10  to  be  added  to  the  upper  ligurc  ;  but  in 
this  case  you  must  add  1  to  the  under  figure  in  the  next  column  before  you 
iiiibtract  iL     This  is  called,  borrowlitir  len.^'' 

PROOF. 

Add  the  rcmaiiider  and  subtrahend  together,  and  if  the  sum  of  them  cor- 
rerpoiid  Willi  the  minuend,  the  work  is  supposed  to  be  right. 

Jlinuend         0     6     5     3         The  numbers  being  Jilaced  with  the  larger 

uppermost,  as  the  rule  directs,  I  begin  \vith  the 
Subtrahend    5     2     7     1     unit  or  right  hand  Iigure   in  the   subtrahend, 

and  say,   ]  from  3  lliorc  remain  2.  which  I  set 

Remainder     3     3     G     2     down,  and  proceeding  to  tens,  or  the  next  figure, 

7  from  5  1  cannot,  I  therefore  borrow,  or  sup- 

Pronf  0     6     5     3     pose  ten  to  be  added  to  the  upper  figure  (5) 

wiiicli  make  15,  ti.en  I  say  7  from  15  and  there  remain  8,  which  I  set  down  ; 
ilu-n  proceeding  to  the  nest  place,  I  sa}',  1  •which  I  borrowed  to  Sis  ^,  and 
3  from  0  and  there  remain  3  ;  this  I  set  down,  and  in  the  next  place  laay  5 
liorn  0  and  tijere  remain  3,  which  1  set  down  and  the  work  is  done. 

PnooF.  I  add  the  remainder  to  the  subtrahend,  and  finding  the  sum  juic 
efj'iul  1o  the  minuend,  suppose  the  work  to  be  right. 

Norn,.  The  reason  of  borroxn-ii'^  ten,  will  appear  i^  we  consider,  that, 
wh".n  iwo  numbers  arc  equally  increased  by  adding  the  same  to  both,  their 
diiTi'.ronce  will  be  equal.  Thus  the  dilTerence  between  3  and  5  is  2  ;  add 
the  niunber  10  to  each  of  these  figures  (3  auil  5)  they  become  13  qnd  15, 
sLiH  the  xlilTerence  is  2.  When  v»e  proceed  as  above  directed,  w7*^add  or 
suppose, to  be  added,  10  to  the  minw.mh  and  we  likewise  add,pi|i^  to  the 
next  higiier  place  of  the  suhiraJhcnd,  which  is  just  equal  iu  value  to  10  of 
I'iC. lower  place. 

5     14     6     5     the  minuend, 
1     2     3     4     2     the  subtrahend. 

Ren?ainder. 


From 
Take 

1 

2 
0 

7 
G 

8 

7 

6     5 

r  3 

3 
G 

rUtrT.  I.  2. 


SIMPLE  SUBTRACTION. 


17 


,Vt  -in  case  of  borrowing  ten,  it  is  a  matter  of  indifference,  as  it  re- 
spects ti.i  operatioH,  whether  we  suppose  ten  to  be  added  to  the  upper 
figure,  and  from  tht  sum  subtract  the  lower  figure  and  set  down  the  dif- 
ference ;  or,  as  21^.  Pike  directs,  first  subtract  the  lower  figure  from  10, 
and  adding  the  difference  to  the  figure  above,  set  down  the  sum  of  this 
difference  and  the  upper  fijrure.  The  latter  method  may  perhaps  be  thought 
more  easy,  but  it  is  conceived,  that  it  does  not  lead  the  understanding  of 
youth  so  directly  into  the  nature  of  the  operation  as  the  former. 

1.  From  10236742317  981062 
Take    8791284506703281 


Rem, 


2.  From 
Take 

Rem. 


10236742317981062 
8  791284506703281 


X  From  21468317012101,  take  668497067382.  Rem.  20899819944719. 


4.  From  3G47 10825193,  take  279403865746. 
6.  From  168012372458,  take   89674807683. 

6.  From  100610528734,  take  99874197867. 

7.  From  628103570126,  take  248976539782. 


Rem.  85306959447. 
Rem.   78337564775. 
Rem.  736330867. 
Rem.  379127030344. 


8.  From  10000,  take  9999.  Rem.  1.    9.  From  10000,  take  1.  Rem.  9999. 

The  distance  of  time  since  any  remarkable  event,  may  be  found  by  sub- 
tracting the  date  thereof  from  the  present  year. 


EXEP.CISFS. 
1.  How  long  since  tbe  Ameri- 
can Independence,   which  was 
declared  in  1 776  ? 
18     17  present  time. 
17     7     6  date  of  Ind. 


Ans.      4     1 


years  since. 


2.  King  Charles,  the  martyr, 
was  beheaded  1648  :  how  many 
yeans  is  it  since  ? 


i 


So,  likewise,  the  distance  of  time  from 
the  occurrence  of  one  thing  to  that  of 
another,  may  be  found  by  subtracting  the 
date  of  the  thing  first  happening,  from 
that  of  the  last. 

EXAMPLE. 

1,  How  long  from  the  discovery  of 
America  by  Columbus,  1492,  to  the  com- 
mencement of  the  war,  1775,  which  gain- 
ed our  Independence  ? 

17     7     6 
14     9     2 


Ans.  2  8  3  years^ 
2.  How  long  from  the  termination  of 
the  war  in  1783,  which  gained  our  Inde- 
pendence, to  the  commencement  of  the 
last  war  between  the  United  States  and 
Great  Britain  in  1812  ?         Ans.  29  ycc.ri 


>\ 


18 


SUPPLEMENT  TO  SUBTRAGTION.  Sect.  {  8. 

SUPPLEMENT  TO   SUBTRACT"-^!.. 


QUESTIONS. 

1.  What  is  Simple  Subtraction  ? 

2.  How  many  numbers  mu«.t  there  be  given  to  perform  that  operation? 
■3.  How  must  the  given  numbers  be  placed  ? 

4.  What  are  thjy  called  ? 

5.  When  the  fi£,'urc  in  the  lower  number  is  greater  than  that  of  the  up- 
per number  from  which  it  is  to  be  taken,  what  is  to  be  done  ? 

6.  How  does  it  appear  that  in  subtracting  a  less  number  from  a  greater, 
the  occasional  borrowing  of  ten  does  not  affect  the  difference  between 
these  two  numbers  ? 

7.  How  is  subtraction  proved  ? 

EXERCISES. 
1.  What  is  the  difference  between         2.  From  a  piece    of  cloth   that 
78360  anJ  5421  ?  measured  691  yards,  there  were  sold 

Ans,  72939.  273  yards  ;  how  many  yards  should 

there  remain  ?  Ans.  418. 


3.  There  are  two  numbers,  whose  4.  What  number  is   that  which 

diflerence  is  375  ;   the  greater  num-  taken  from  176  leaves  96  ? 

ber  is  862  ;  I  demand  the  less  ?  Ans.  79. 
Ans.  487. 


V 


5.  Suppose  a  man  to  have  been  born  in  the  year  1745,  how  old  was  he 

in  1799?  Ans.  54  years. 


6.  What  number  is  that  to  which  if  you  add  789  it  will  become  6350  ? 

Ans.  6561. 


in 


7.  Supposing  a  man  to  have  been  63  years  old  in  the  year   1801 
what  year  wijij^born  ?  Ans.  in  the  year  1738. 

8.  At  the  census  in  1800,  the  number  of  inhabitante  in  the  New-England 
States  was  12^3011  ;  at  the  late  census  in  1810,  the  number  was  1471937;  s 
What  was  the  ifcre;u3e  of  th'*  ponuIatioD  ia  the  New-Eng-lantl  States  in  the  : 
'on  years  between  lOOOand  JfJlO?  ''-~  '"^'^926 


Sect.  I.  3. 


SIMPLE  MULTIPLICATION. 


19 


i  3.  SIMPLE  MULTIPLICATION. 


Simple  Multiplication  teaches,  having  two  numbers  given  of  the  same 
denomination,  to  find  a  third  which  shall  contain  either  of  the  two  given 
cumbers  as  niafiy  times  as  the  other  contains  a  unit.  Thus,  8  multipUed  by 
5,  or  5  timfes  8  is  40 — The  given  numbers  (8  and  5)  spoken  of  together,  are 
called  Factors.  Spoken  of  separately,  the  first  or  largest  number  (8)  or 
number  to  be  multiplied,  is  called  the  Multiplicand;  the  less  Jjujnber  (5) 
or  number  to  multiply  by,  is  called  tho- multiplier,  and  the  amount  (40)  the 
Product. 

Before  any  progress  can  be  made  in  this  rule,  the  following  Table  must 
be  committed  perfectly  to  memory. 


MULTIPLICATION 

TABLE. 

H 

2 

~~3 

4|  5 

6 

7 

8 

9 

10  1 

11 

r: 

2| 

4 

e 

8  1  10 

12 

14 

16 

18 

20  i 

22 

24 

3| 

6 

9 

12|15 

18 

21 

24 

27 

30  1 

33 

36 

4| 

8 

12 

IG  1  20 

24 

i  28 

32 

36 

40 

44 

48 

51 

10 

lo 

20  1  25 

30 

i  35 

40 

45 

50  1 

56 

60 

C| 

12 

18 

24  1  3(» 

36 

42 

48 

54 

60  1 

66 

72 

7| 

14 

21 

28  1  3r> 

42 

!  41) 

56 

63 

70 

77 

84 

M 

16 

24 

32  1  40 

43 

56 

64 

72 

80  1 

88 

96 

•^1 

18 

-  • 

36  ,  46 

51 

i  f^2 

7'.'. 

81 

90  j 

99 

108 

10  i 

20 

30 

40  1  5(.i 

GO 

!  70 

80 

90 

100  1 

no 

120 

HI 

22 

33 

44  1  65 

C6 

1  77 

88 

99 

110  i 

121 

132 

12  j 

24 

36 

18  I  60 

72 

84 

96 

108 

120  1 

132 

144 

1/ 


By  this  table  the  |>roduct  of  any  two  figures  will  be  found  in  that  square 
which  is  on  a  line  ynttk  thfe  one  and  directly  under  the  other.  Thus,  56 
the  product  of  7  and  8,  will  be  found  on  a  Une  with  7  and  under  8  :  so  2 
times  2  is  4  ;  3  times  3  is  9,  &c. — In  this  way  the  table  must  be  learned 
and  remembered.    ^ 

RULE. 

1.  Flare  the  numbers  as  in  Subtraction,  the  larger  number  uppermost 
with  units  under  units,  o:c.  and  then  draw  a  line  below. 

2.  Ulien  the  multiplier  does  not  exceed  12  :  begin  at  the  right  hand  of  the 
tnuUiplicand,  and  multiply  each  figure  contained  in  it  by  the  multiplier,  set 
ling  down  all  over  even  tens  and  carrying  a-  in  addition^^^. 

3.  Wlten  the  multiplier  exceeds  12  ;  multiply  by  e»<fvl|gurc  separately, 
fu'st  by  the  ufiits  of  the  mtilfiplior,   as  directed  above, ^ften   by   the   ienii, 

S"  ltd  the  other  figures  in  their  order,  remembering  always  to  place  the  first 
giire  of  each  product  directly  under  the  figure  by  which  you  multiply  ; 
aving  gone  through  in  this  manner  with  each  figure  in  the  multiplier,  add 
lieir  several  products  together,  and  the  sum  of  them  will  be  the  prodnct 
iquired. 


20  SIMPLE  MULTIPLICATION.  ,     Sect.  I.  3. 

EXAMPLES. 

1.  Multiply  5291  by  3.  The  numbers  being  placed  as  seen  under 

OFERATioN.  the  operation,  say — 3  times  1  is  3  ;  which 

5  2  9  1   Multiplicand.       set   down  directly  under  the   multipher  ; 

3  Multiplier.  then  3  times  9  is  27  ;  set  down  7  and  carry 

— — 2.     Again  3  times  2  is  6  and  2  I  carry  is  8  ; 

15  8  7  3  Product.  set  down  8 :  then  lastly,  3  times  5  is  15, 

which  set  down,  and  the  work  is  done. 

2.  What  is  the  product  of  4175  Multiplied  by  37  ? 

Place  the  Factors  thus,    M     ^     ^    ^  m"!!'?!*"""*^- 
'5  3     7  Multipher. 

2     9     2    2     5  Prod,  by  the  unit  (7)  of  the  mul- 
1     2     5    2     5        Product  by  the  tens  (3)     [tipUer. 

15    4     4     7     5  Product  or  answer.  % 


fn  this  cxanaple,  as  the  multipher  exceeds  12,  therefore  you  must  multi- 
ply by  each  figure  separately.  First,  by  the  units  (7)  just  in  the  manner 
vi'  the  other  example.  Secondly  by  the  tens  (3)  in  the  same  way  except- 
ing only,  that  the  first  figure  of  the  product  in  the  multiplication  by  3,  must 
be  placed  under  the  3,  that  is,  under  the  figure  by  which  you  multiply. 

Lastly,  add  these  two  products  together,  and  the  sum  of  them  is  the  an- 


Prook. — The  better  way  of  proving  Multiplication  is  by  Division  :  but 
till  the  pupil  shall  have  been  instructed  in  that  rule  he  may  make  use  of 
the  following 

METHOD. 

Cast  the  nines  out  of  {.nw"  Multiplicand,  and  set  the  remainder  at  the 
right  hand  of  a  cross;  do  the  same  with  the  Multiplier  and  set  the  remain- 
der at  the  left  hand  of  the  cross  ;  then  multiply  the  figures  at  the  right  and 
left  of  the  cross  together,  cast  the  nines  out  of  the  product,  and  set  the  re- 
mainder over  the  cross ;  also  cast  the  nines  out  of  the  answer  or  product 
of  the  multiplicand  and  multiplier,  and  set  the  remainder  under  the  cross, 
which  will  be  the  same  as  that  over  it  if  the  work  be  right. 

NoTK. — To  east  ''ic  ninea  nut  of  any  number,  proceed  tiius:  beginning  at  the  right  hand 
of  the  nuinhfr,  acid  rtie  figures ;  when  the  sum  exceeds  9,  drop  the  sum  and  begir;  anew, 
l)y  addiiiii,  first,  the  figures  wliich  would  express  it.     Pass  by  the  nines,  and  when  the  sum 
comes  out  exactly  y.^eglect  it ;  what  remains  after  tlie  last  addition,  will  be  the  renaain- 
dfcr  sought:  for  oiknpie — suppose  it  be  required  to  cast  the  9's  out  of  576394,  proctied 
thus : — .'.  to  7  is  1'2,  whicli  sum  (luelre)  as  it  exceeds  9  you  must  drop,  and  beginning  anert', 
first  add  the  figures  (12)  which  would  express  twelve,  saying  1  to  2  is  3  (proceeding  wiilh 
the  other  figures  wliich  remain  to  be  added)  and  6  are  9,  being  eiOf//v  inne,  neglect  it,  p  i  .J 
lieniii  again;  3  to  9  ai-e  twelve;  again  drop  the  sum  (twelre)  and  add  the  figures  (4]|h 
vhich  would  express  it,  1  to  2  is  3  and  4  are  7,  which  sum  (7)  is  the  remainder  after  -^Wk 
last  addition,  or  tJie  thing  sought,  iuid  is  the  remaiilder  tltat  would  be  left  after  dividing  tl?.    ] 
sum  57o;is>4  by  U.  ■'-■     ' 


Sect.  I.  3.  SIMPLE  MULTIPLICATION.  fl 

3.  Multiply  7  6  5  3  0  2  Multiplicand. 
6  5  Multiplier. 

PROOF. 

3  8  2  6  6  10  1 

4  5  9   18   12  2X5 


1 


49744630  Product. 


Proof.  I  cast  the  9's  out  of  the  Multiplicand  and  set  the  cemauider  (6) 
at  the  right  hand  of  the  cross  ;  I  do  the  same  with  the  MuUrplier,  and  set 
the  remainder  (2)  at  the  left  hand  of  the  cross  ;  the^e  remainders  I  multi- 
ply together,  and  casting  out  the  9's  fi^m  the  product  (10)  the  remainder 
is  1,  which  1  set  at  the  top  of  the  cross  ;  I  then  cast  out  the  9's  from  the 
Product  (49744630)  and  set  the  remainder  (1)  at  the  bottom  of  the  cross, 
which  as  it  agrees  with  the  remainder  at  top,  I  suppose  the  work  to  be 
right. 

There  is  nothing  more  easy  than  proving  Multiplication  by  this  method 
•o  soon  as  the  scholar  shall  have  giveo  it  such  attention,  as  to  make  it  a 
little  familiar. 

Note.  Should  the  Multiplier  or  Multiplicand,  either  or  both,  be  less 
than  9,  they  are  to  be  taken  as  the  remainders. 

4.  Multiply  37846  by  235.     Pro-         5.  Multiply  14356  by  648.     Pro- 
duct, 8893810.  duct,  9302688. 


6.  Multiply  29831  by  952.     Pro-        7.  Multiply  93966  by  8704.    Pro- 
duct, 28399112.  duct,  817793024. 


22  SIMPLE  MULTIPLICATION.  Sect.  I.  3. 

8.  Multiply  98704563  by  75604     Prod.    746245978105? 

9.  -  3462321-      96484--  334058570364 

10.  -  A27535-15728        -  8297070480 

11.  -  2758^7-19725         -  5440687575 

12.  -  696374-463957        -  323087591918 

Contractions  and  varieties  in  Multiplication. 

Any  number  which  may  be  produced  by  the  multiphcation  of  two  or 
mpre  numbers,  is  called  a  composite  number.  Thui*  15  wuicli  arises  from 
the  multiplication  of  5  and  3,  (3  times  5  is  15)  is  a  composite  uumber  ;  and 
these  numbers,  6,  and  3,  are  called  component  parts.     Therefore. 

1.  If  the  multiplier  be  a  composite  number ;  multiply  first  by  one",  of  the 
component  parts,  and  that  product  by  the  other  ;  the  last  product  will  be 
the  answer  sought. 

EXAMPLES. 
1.  Multiply  6  7  by  1  5 

OPERATION. 
6    7 

5  one  of  the  component  parts. 


3  3  6 

3  the  other  component  part. 


10  0  5  Product  of  67  mult,  by  15. 
2.  Multiply  367  by  48,  Product,  17616. 


OPERATION. 

Consider  first  whut  two  numbers  mu- 
tiplied  together  will  produce  48  ;  that  is, 
what  arc  the  component  parts  oi  48  ? — 
Answer,  G  and  8  (6  times  0  is  -IC)  therer 
fore  multiply  367  first  by  one  of  the  com- 
ponent parts,  and  the  product  thence 
arising  by  the  other ;  the  last  product 
will  be  the  answer  sought. 


3.  Mult.  583  by  66.    Prod.  32648.     4.  Mult.  1086  by  72.  Prod.  78192. 

OPERATION.  *  OPERATION. 


2.  "  When  there  are  cyphers  on  the  right  hand  of  either  tha  multiplicand  or 
*'  multiplier,  or  both,  neglect  those  cyphers ;  then  i)lace  the  significant  fig- 
"  ures  under  one  another,  and  multiply  by  them  only  ;  add  them  together 
"  as  before  directed,  and  place  to  the  right  hapd  as  mcaiy  cyphers  as  there 
"  are  in  both  the  factors." 


Sect.  I.  3. 


SIMPLE  MULTIPLICATION. 


S3 


EXAMPLES. 


I.  Multiply  65430  by  5200. 

OPERATION. 

6     5     4      3     0 

5     2     0     0 


Here  in  the  multiplication  of  65430  by 
5200,  the  cyphers  are  seen  neglected, 
and  regard  paid  only  to  the  significant 
figures.  To  the  product  are  annexed 
3  cyphers  ;  equal  to  the  number  of  cy- 
phers neglected  in  the  factors. 


340236000 


i.  Mult.     3     6     5    0    0 
By  7     3     0 


3.  Mult.  78000  by  600. 

Product^  46800000. 


Prod.  2    6     6    4    6    0    0    0 


3.  When  there  are  cyphers  between  the  significant  figures  of  the  multiplier, 
omit  the  cyphers  a^d  multiply  by  the  significant  figures  only,  placing  the 
first  figure  of  each  product  directly  under  the  figure  by  which  you  multi- 
ply, and  adding  the  products  together,  the  sum  of  them  will  be  the  product 
of  the  given  numbers. 

EXAMPLES. 


J.  Mult.  154326  by  3007. 

OPERATION. 

15  4  3  2  6 
3  0  0  7 


10  8  0  2  8  2 
4  6  2  9  7  8 

464058282 


In  this  example,  the  cyphers  in  the  mul- 
tiplier are  neglected,  and  154326  multi- 
plied only  by  7  and  by  3,  taking  care  to 
place  the  figure  in  each  product  directly 
under  the  figure  from  which  it  was  ob« 
tained. 

2. 
3     4     6     7 
3     0     2 


10     4     4     0     14 


U  SIMPLE  MULTIPLICATION.  S«ct.  I.  3; 


48976850 
4     0     0     0     3     0 


19592209305500 

4.  When  the  Miikiplier  is  9,  99,  or  any  number  of  9's,  annex  as  many  cy- 
pher's to  the  Multiplicand,  and  from  the  number  thus  produced,  subtract 
the  multiplicand,  the  remainder  will  be  the  product. 

EXMIPLES. 

1.  Mult.  6547  by  999  Write  down  the  Multiplicand,  place  as 

OPERATION.  many  cyphers  at  the  right  hand  a^  there 

6  5  4  7  0  0  0  are  9's  in  the  Multiplier  for  a  minuend  ; 

6  5  4  7  underneath  write  again  the  multiplicand  for 

■  a  Subtrahend,  subtract,  and  the  remainder 

6  6  4  0  4  6  3  is  the  product  of  6547  multiplied  by  999. 

2.  3. 

^^99  (  ^'■'''^"^^  640827  "^^^J  i  PrfidpCt,  695079 


4.      " 
$364976 
9    9    9    9 


i  Preduct,  63844375024 


Sect.  I.  3.      SUPPLEMENT  TO  MULTIPLICATION.  25 

SUPPLEMENT  TO  MULTIPLICATION. 


\ 


QUESTIONS. 

1.  What  is  Simple  Multiplication  ? 

2.  How  many  numbers  are  required  to  perform  that  operation  ? 

3.  Collectively  or  together,  what  are  the  given  numbers  called  ? 

4.  Separately,  what  are  they  called  ? 

5.  What  is  the  result,  or  number  sought,  called  ? 

6.  In  what  order  must  the  given  numbers  be  placed  for  multiplication  ? 

7.  How  do  you  proceed  when  the  multiplier  is  less  than  12  ? 

8.  When  the  multipHer  exceeds  12  what  is  the  method  of  procedure? 

9.  What  is  a  composite  number  ? 

10.  What  is  to  be  understood  by  the  component  parts  of  any  number  ? 

11.  How  do  you  proceed  when  the  multiplier  is  a  composite  number? 

12.  When  there  are  cyphers  on  the  right  hand  of  the  multiplier^  multi- 
plicand, either  or  both,  what  is  to  be  done  ? 

13.  When  there  are  cyphers  between  the  significant  figures  of  the  mul- 
tiplier, how  are  they  to  be  treated  ? 

14.  When  the  multiplier  consists  of  9's  how  may  the  •peration  be  con* 
tracted  ? 

15.  How  is  Multiplication  proved  ? 

16.  By  what  method  do  you  proceed  in  casting  out  Lie  y's  from  any 
number  ? 

17.  How  is  Multiphcation  proved  by  casting  out  the  9's  ? 

EXERCISES. 

1.  What  sura  of  money  must         Note.    The  scholar's  business   in  all 

be  divided  between  27  men,  so     questions  for  Arithmetical    operations,   is 

that  each   may   receive    115     wholly  with  the    numbers    given;    these 

dollars.  .fins.  3105.         are  never  less  than   two  ;   they  may  be 

more,  and  these  numbers  in  one  way  or 
another,  are  always  to  be  made  use  of  to 
find  the  answer.  To  these,  therefore,  he 
must  direct  his  attention,  and  carefully 
consider  what  is  proposed  by  the  question 
to  br  known. 
D 


2C 


SUPPLEMENT  TO  MULTIPLICATION.      Sect.  I.  3. 


2.  An  army  of  10700  mcD  having 
plundered  a  city,  took  so  much 
money,  that  .when  it  was  shared 
among  them,  each  man  received  46 
dollars  ;  what  was  tlie  sum  of  money 
taken  ?  Ans.  492200. 


3.  There  were  175  men  employed 
to  finish  a  piece  of  work,  for  which 
each  man  was  to  receive  13 dollars; 
what  did  they  all  receive  ? 

Ans.  2276. 


4.  .Suppose  a  certam  town  coii- 
tiiins  145  houses,  each  house  two 
families,  and  each  family  6  inhabit- 
ants ;  how  many  would  be  the  inhab- 
itante  of  that  town  ?         Jlns.  1740. 


5.  If  a  man  earn  2  dollars  per 
week,  how  much  will  he<»^n  in 
5  years,  there  being  62  weeks  in  a 
year  ?  Ans.  620  dolls. 


6.  How  much  wheat  will  36  men 
thrash  in  37  days,  at  5  bushels  per 
day  each  man  ? 

Ans.  6660  bushels. 


7.  If  the  price  of  wheat  be  1  dol- 
lar per  bushel,  and  4  bushels  of 
wheat  make  1  barrel  of  flour,  what 
will  be  the  price  of  175  barrels  of 
flour  ?  Ans.  700  dolls. 


I 


Sect.  I.  4.  SIMPLE  DIVISION.  27 

«  4.  SIMPLE  DIVISION. 


SIMPLE  DIVISION  teaches,  having  two  numbers  given  of  the  same 
denomination,  to  find  how  many  times  one  of  the  given  numbers  contains 
the  other.  Thus,  it  may  be  required  to  know  how  many  times  21  contains  7  ; 
the  answer  is  3  times.  The  larger  number  (21)  or  number  to  be  divided, 
is  called  the  Dividend ;  the  lesser  number  (7)  or  number  to  divide  by,  is 
Cj^led  the  Divisor;  and  the  answer  obtained,  (3)  the  Qttotient. 

After  the  operation,  should  there  be  any  thing  left  of  the  Dividend,  it  is 
called  the  Remainder.  This  part,  however,  is  uncertain  ;  sometimes  there 
'\s  no  remainder.  When  it  does  happen  it  will  always  be  less  than  the  di- 
visor, if  the  work  be  right,  and  of  the  same  name  with  the  dividend. 

RULE. 

1.  "  Assume  as  many  figures  on  the  left  hand  of  the  dividend  as  contain 
"  the  divisor  once  or  ollenor  ;  find  how  many  times  they  contain  it,  and 
"  place  the  answer  as  the  highest  figure  of  the  quotient. 

.2.  ^'  Multiply  tjie  divisor  by  the  figure  you  have  foupd,  and  place  the 
"  product  under  that  part  of  the  dividend  from  which  it  was  obtained. 

3.  •'  Subtract  the  product  from  the  figures  above  it. 

4.  "  Bring  down  the  next  figure  of  the  dividend  to  the  remainder  and 
■'  divide  the  number  it  makes  up  as  before." 

When  you  have  brought  down  a  figure  to  the  remainder,  if  the  number 
it  makes  up  be  still  less  than  the  divisor,  a  cypher  must  be  placed  in  the 
quotient,  and  another  figure  brought  down. 

EXAMPLES. 
I.   Divide  127  by  5. 
Divisor.     Dividend.     Quotient.  The  parts  in  Division  are  to  stand 

"))      1     2     7     (2     6  thus,  the  dividend  in  the  middle,  the 

1     0  divisor  on  the  left  hand,  the  quotient 

on  the  right,  with  a  half  parenthesis 

2     7  separating  them  from  the  dividend. 

2     5 


2     Remainder. 

Proceed  in  this  operation  thus-i-It  being  evident  that  the  divisor  (5) 
cannot  be  cotitaiticd  in  the  first  figure  (1)  of  the  dividend,  therefore  assume 
the  two  first  fiii^uros  (12)  and  inquire  how  often  5  is  contained  in  12  ;  find- 
ing it  to  be  2  limes,  place  2  in  tlie  quotient,  and  multiply  the  divisor  by 
it,  saying  2  times  "i  is  10,  and  jdace  tlie  sum  (10)  directij'  under  12  in  the 
dividend.  Subti  s't  10  from  12  and  to  the  remainder  (2)  bring  down  the 
next  tigure  (7)  at  the  right  hand,  making  with  the  remainder  27.  Agam 
inquire  how  >nauy  times  6  in  27  ;  5  times  ;  place  5  in  t)ie  tjuotient,  multiply 
the  divisor  (6)  by  the  last  quotient  figure  (o)  saying  5  times  5  is  26,  place 
the  sum  (25)  under  27,  subtract  and  the  v.ork  is  done.  lience  it  appears 
that  127  contains  5,  25  times,  with  a  remainder  of  2,  which  was  left  after 
the  last  subtraction.  "  V 

This  Rule,  perhaps  at  first  will  appear  Intricate  to  the  young  student, 

although  it  is   attended  with    no    dilhcuUy.      His  liability  to  errors  will 

chielly  arise  from  the  diversity  of  proceedings.     To  assist  his  recollection, 

let  him  notice  that  ^  1.  Find  how  many  times,  Lc. 

rri        X  rn-   •  •  r  ; -•   Multiply. 

1  he  step?:  of  Divi':i'>n  arc  Jour  <  ,,    c  i »     "! 
^  ^  ••    Subtract. 

Turing  down- 


28  SIMPLE  DIVISION.  S«ct. 


h  s 


It  is  sometimes  practised  to  malce  a  point  (.)  under  the  figures  in  the'j 
dividend,  as  they  are  brought  down,  in  prdcr  to  prevent  mistakes.  \ 

When  the  divisor  is  a  large  num1>er,  it  cannot  always  certainly  be  kfu>wn  i 
how  many  times  it  may  be  taken  in  the  figures  wbicn  are  assumed  p;:'tUdj 
left  hand  of  the  dividend  till  after  the  first  steps  in  division  axe  goii^  ever, 
but  the  learner  must  try  so  many  times  as  his  judgment  may  best  dicute, 
•and  after  h(j  has  raultipUed,  if  the  product  be  greater  than  the  number  as*  - 
sumed,  or  that'  number  in  which  the  divisor  is  taken,  then  it  may  alwayg 
be  kpowp  that  the  quotient  figure  is  too  large  ;  if  after  he  has  muhiplied 
and  subtracted,  the  remainder  be  greater  than  the  divisor,  then  the  quotient 
figure  is  not  large  enough,  he  must  then  suppose  a  greater  number  of 
time?,  and  proceed  again.     This  at  first  may  occasion  some  pezT)lexity,  but  ' 
the  attfjntive  learner  after  some  practice,  will  generally  hit  on  the  right  j 
II  umbo  r. 

2.  Let  it  be  required  to  divide  7012  by  62. 

OPERATION. 

Divisor.     Dividend.  Q,uotient.         In  this  operation  it  is  left  for  the  pchol- 

»  2)  7  0  1   2  (I  3  4               ar  to  trace  the  steps  of  procedure  without 

5  2  having  them  particularly  pointed  otit  to 

him  by  words. 

18  1 
1  5  6 


2  5  2 
8  0  8 

4  4  Remainder. 

PROOF. 

Division  may  be  proved  by  multiplication.  v 

RULE. 
"  Multiply  the  Divisor  and  Quotient  together,  and  add  the  remainder,  if  j 
"  there  be  any  to  the  product ;  if  the  work  be  right,  the  sum  will  be  equal 
•'  to  the  dividend." 


Take  the  last  Example. 
The^Quotieat  was    1  3  4  |  ..^j^.^j^  ^,^^  ^^^^^^^^^ 


2  6  8 
6  7  0 

4  4  Remainder  added. 


7  0  12  Equal  to  the  dividend. 
Another  and  more  expeditious  way  of  proving  Division  is 

By  casting  out  the  9's.  | 

Cast  out  the  9's  from  the  Divisor  and  the  Quotient,  multiply  the  resulte, 
and  to  the  product,  add  the  remainder  if  any  after  division  ;  from  the  sum 
of  the^e  cast  out  the  9's,  also  ca.st  out  the  9's  from  the  Dividend,  and  if  the 
two  last  results  agree,  the  work  is  supposed  to  be  right. 


I^KCT.  I.  4.  SIMPLE  DIVISION.  59 

3.  Divide  17364  by  86. 

OPERATION.  PROOF. 

Divis.     Divid.     Quot.    9's  oiii  of  (Divis.)  86  Rem.  5  >  Multiplied 
86)1  7  3  5  4(2  0  1 {Quot.)    201  Rem.  3  J  together. 


1   7  2 


16 


_^,^  15  4  Remainder  68  added. 


r..- 


8  6  — 

9'9  out  of  83  Rem.  2  }  agreeing 

6  8  Rem.         9*8  out  of  {Divid.)   17334  Rem.  2  J  togetWr 

4.  Divide  153598  by  29.        quotient  5296.     Rem.  14. 

6  Divide  8893810  by  235.    Qwr.  37846. 


6.  Divide  30114  by  63.    (Quotient  418. 

7.  Divide  9302638  by  648.     quot.  14356. 


30  SIMPLE  DIVISION.  8fct.  I. 

8  Divide  974932  by  365.     Quotient  2671.     Rem.  17. 


ii 


9.  Divide  5221580  by  68705.     Ouot.  7t 


10.  Di>id*»  3228242  dollars  equaUy  among  563 men;  how  many  dollars 
must  each  man  receiye  ?  Jlns.  6734. 

From  a  view  of  the  question,  it  is 
evident,  that  the   dollars   must  be 
divided  into  as  many  parts  as  there 
^^  are  men  to  receive   them;  conse- 

'^'  quently,  the  number  of  dollars  must 

be  made  the  dividend,  and  the  num- 
ber of  men  the  divisor;  the  quotient 
will  then  show  how  many  dollars 
each  man  must  receive. 

1 1.  How  manv  limes  does  1030603Gk". 
contain  3215?  .'3ns.  320561  times 


Sect.  I.  4. 


SIMPLE  DIVISION. 


31 


Contractions  and  varieties  in  Division. 

I.  When  the  divisor  does  not  exceed  12,  the  operation  may  be  performed 
without  setting  down  any  ligures  excepting  the  quotient,  by  carrying  the 
computation  in  the  mind.  The  units  which  would  remain  alier  subtractins^ 
the  product  of  the  quotient  figure  and  the  divisor  Irom  the  ligures  assumeci 
of  the  dividend,  mnst  be  accounted  so  many  tens,  and  be  «i'jj)posed  to  stand 
at  the  left  hand  of  the  next  figure  in  the  dividend,  then  consider  again  how 
often  the  divisor  may  be  had  in  the  sum  of  them.  Proceed  in  this  waj'  till  ali 
the  figures  in  the  dividend  have  been  divided.    This  is  called  short  division. 

EXAMPLES. 

1.  Divide  732  by  3. 

OPERATION. 

3)732  Here  I  say,  how  often  3  in  7,  knowing 

it  to  be  2  times,  I  place  2  in  the  quotient, 

2  4  4  Q^uotient.  then  considering  that  the  quotient  figure 

(2)  and  the  divisor  (3)  multiplied  to- 
gether would  be  6,  and  that  this  product 
(6)  subtracted  from  7  in  the  dividend,  would  leave  1,  I  then  consider  tl)i.> 
remainder  (1)  as  standing  at  the  lefl  hand  of  the  next  figure  (3)  of  the  divi- 
dend, which  together  make  13.  i  now  say  how  many  times  3  in  13 — I 
times,  therefore  I  place  4  in  the  quotient,  which  multiplied  into  the  divisor 
(3)  would  be  12,  and  12  subtracted  from  13  would  leave  1,  which  consid- 
ered as  standing  at  the  left  Rand  of  the  next  or  last  figure  (2)  of  the  divi- 
dend would  make  12  ;  again,  how  many  times  3  in  12 — 4  times — I  then 
place  4  in  the  quotient,  which  multiplied  into  the  divisor  (3)  is  12,  this 
product  (12)  I  consider  as  subtracted  from  12,  I  find  there  will  be  no  re- 
mainder, and  the  work  is  done. 

2.  Divide  37426  by  7. 

OPERATION. 

7)37426 


^uot.    5  3  4  6  Rem.  4 


Here  I  say  how  often  7  in  37  ?  5 
times  and  two  remain  ;  then  how 
often  7  in  24  ?  3  times  and  3  re- 
main ;  how  often  7  in  32  ?  4  times 
and  4  remain  ;  lastly,  how  often  7  in 
46  ?  6  times,  4  remain. 
Cluot.  77664505790 
124215042924 
3.6537338604 

-  38306588315 

-  60702378652 


Rem.  2 

-  -    1 

-  -    4 

-  -    7 


3.  Divide  310658023162  by  4 

4.  -  -  -  621075214621   -    5 

5.  -  -  -  213524031628  -    6 

6.  -  -  -  306452706527  -    8 

7.  -  -  -  546321406968  -     9 
II.   JVhen  there  are  cyphers  at  the  right  hand  of  the  divisor,  cut  them  off, 

also  cut  off  an  equal  number  of  figures  from  the  right  hand  of  the  dividend 
and  place  these  figures  at  the  right  hand  of  the  remainder. 

EXAMPLES. 
1.  Divide  6203946  by  5700. 


OPFRATION. 

57  I  00)62039  |  46(1088 
57 . . . 


603 
456 


479 
456 

2346 


Here  are  two  cyj^hers  on  the  riglit 
hand  of  the  divisor  which  I  cut  ofl", 
also  I  cut  off  two  figures  (4G)  from 
the  dividend  and  to  the  right  hand  of 
the  re'mainder  after  the  last  divisiou 
(23)  I  place  the  figures  cut  otT  from 
the  dividend  (40)  which  make  the 
whole  rr»maindfr  ?346. 


32  SIMPLE  DIVISION.  Sect.  I.  4 

2.  Divide  379432  by  6500    q,uot.  68.     Rem.  2432. 


3.  Divide  2r64303721  by  83000.     Quot.  33307.    Rem.  22721. 


4.  J0ie7i  the  divisor  is  10,  100,  1000,  or  1,  ti'j'^A  any  number  of  cyphers  an- 
nexed, cut  oif  as  many  figures  on  the  right  hand  of  the  dividend  as  there 
are  cypliers  in  the  divisor  j  the  figures  vehich  remain  of  the  dividend  coin- 
pose  tlie  quotient ;  those  cat  off,  the  remainder. 

EXAMPLES. 

1.  Divide  1576  by  10.  Here  Ave  have  one  cypher  in  the  divi-    j 

OPERATION.  sor,  therefore  cut  off  one  figure  (6)  from    ' 

1  I  0)   1   5  7  I  6  the  dividend;  what  remains  (157)  is  the 

quotient,  and  the  figure  cut  off  (6)  the 
remainder. 

2.  Divide  3217  by  100.     Q^uot.  32.  Rem.  17. 

?y.  Divide  76421795  by  1000. 

Qvot.  76421,  Rem.  795 


8tcT.  I.  4.  SUPPLEMENT  TO  DIVISION.  33 

SUPPLEMENT  TO  DIVISION. 


QUESTIONS. 

1.  What  is  Simple  Division  ? 

2.  How  many  numbers  must  there  be  given  to  perform  the  operation  ? 

3.  What  are  the  given  numbers  called  ?  f 

4.  How  are  they  to  stand  for  Division  ?  • 
'5.  How  many  steps  are  there  in  Division  ? 

6.  What  is  the  first  ?  the  second  ?  the  third  ?  the  fourth  ? 

7.  What  is  the  result  or  answer  called  ? 

8.  Is  there  any  other  or  uncertain  part  pertaining  to  Division  ?    What  is 

it  called  ? 

9.  Of  what  name  or  kind  is  the  remainder  ? 

10.  What  is  short  Division  ? 

1 1.  When  there  are  cyphers  at  the  right  hand  of  the  divisor,  what  is  to  be 

done  ? 

12.  What  do  you  do  with  figures  cut  off  from  the  dividend  when  there  are 

cyphers  cut  off  from  the  divisor  ? 

13.  When  the  divisor  is  10,  100,  or  1  with  any  number  of  cyphers  annex- 

ed, how  may  the  operation  be  contracted  ? 
'14.  How  many  ways  may  Division  be  proved  ? 

15.  How  is  Division  proved  by  Multiplication  ? 

16.  How  may  Division  be  proved  by  casting  out  the  9's  ? 

EXERCISES. 

1.  Suppose  an  estate  of  36582  dol-  2.  An  army  of  15000  men  having 

lars  to  be  divided  among  13  sons,  how  plundered  a  city,  and  took  2G25000 

much  would  each  one  receive  ?  dollars,  what  was  each  man's  share  ' 

Jlns.  2814  dolls.  .Sns.  175  dolU. 

E 


34  SUPPLEMENT  TO  DIVISION.  Sect.  I.  4» 

3.  A  certain  number  of  men  were  4.  If  7412  eggs  be  packed  ia  34 

concerned  in  the  payment  of  ^18950  casks,  how  many  in  a  cask  ? 

and  each  man  paid  25  dollars,  what  Ans.  218. 
was  the  number  of  men  ?    Ans.  768. 


I 


5.  A  farm  of  375  acres  is  let  for  6.  A  field  of  27  acres  produces 
i  125  dollars  ;  how  much  does  it  pay  675  bushels  of  wheat ;  how  much  is 
per  acre  ?  Ans.  3  dolls.  that  per  acre  ?        Ms.  25  bushels. 


7.  Supposing  a  man's  income  to  8.  What  number  must  I  multiply 

be  2555  dollars  a  year  ;  how  much  by  13,  that  the«product  may  be  871  ? 

is  that  per  day,  there  being  365  days  ^ns.  67. 
in  a  year.          ,              Ans.  7  dolls. 


Sect.  I.  5.  COMPOUND  ADDITION.  S5 

J  5.  COMPOUND  ADDITION. 


COMPOUND  ADDITION  is  the  adding  of  numbers  which  consist  of 
articles  of  different  value,  as  pounds,  shillings,  pence,  and  farthings,  called 
different  denominations ;  the  operations  are  to  he  regulated  h^j  the  value  of 
the  articles,  which  must  be  learned  from  the  Tables. 

RULE    FOR    COMPOUND    A»DITIO?f. 

1.  Place  the  numbers  so  that  those  of  the  same  denomination  may  stand 
directly  under  each  other. 

2.  Add  the  first  column  or  denomination  together,  and  carry  for  that 
number  which  it  takes  of  the  same  denomination  to  make  1  of  the  next 
higher.  Proceed  in  this  manner  with  all  the  columns,  till  you  come  to  the 
last,  which  must  be  added,  as  in  Simple  Addition. 


1.  OF  MONEY. 

TABLE. 

4  Farthings  qr.      \  L  Penny,  marked  d. 

12  Pence                  >    make  one    /.Shilling,  s. 

20  Shillings              )                         (Pound,  £. 

EXAMPLES. 

1.  What  is  the  sum  of  £61.     17.?.    5d. £13.    3s.  8c/. and  of 

£5.     16s.     llrf.  when  added  together  ? 

OPERATION. 

£.         s.  d. 


^t.         *o         q(      Those  numbers  of  ^the  same  denomination  placed 


61  17 

.p       ."  ^  under  each  other,  .as  the  rule  direct? 


80  18         0 

I  begin  with  the  right  hand  column  or  that  of  pence,  and  having  added 
it,  find  the  sum  of  the  numbers  therein  contained  to  be  24  ;  now  as  12  of 
this  denomination  make  one  of  the  next  higher,  or  in  other  words  12  penC'i 
make  one  shilling,  therefore  in  this  or  in  the  column  of  pence  I  must  can  y 
for  12  ;  I  now  inquire  how  often  12  is  contained  in  24,  the  sam  of  the  fii-t 
column  or  that  of  pence  ;  knowing  it  to  be  2  times  and  nothing  over,  I  set 
down  0  under  the  column  of  pence,  and  carry  2  to  that  of  shillings,  to  be 
added  into  the  second  column,  saying,  2  I  carry  to  6  are  8,  and  3  are  1 1 , 
and  7  are  18,  and  10  to  18  are  28,  and  ten  again  are  38  (for  so  each  figure 
in  ten's  place  must  be  reckoned,  1  in  that  place,  being  equal  in  value  to  10 
units.)  Now  as  20  shillings  make  one  pound,  therefore  in  the  column  of 
shillings,  I  carry  for  20  ;  I  then  inquire  how  ofi:en  20  in  38  '.'  cnce,  and  13 
remains  ;  therefore,  I  set  down  directly  under  the  column  of  shilHiigs  18, 
what  38  contains  more  than  20,  and  for  the  even  20  carry  1  to  pounds  oi? 
the  last  column,  which  is  to  be  added  alter  the  manner  of  Simple  Addiiion. 

JVote. — The  method  of  proof  for  Compound  Addition  is  the  same  as  ta<ir 
of -Simple  Addition. 


/ 


Z5 


COMPOUND  ADDITION. 


Skct.  f.  %. 


X 


£. 

s. 

d. 

9r. 

£. 

s. 

d. 

18 

4 

11 

1 

371 

15 

6 

26 

15 

3 

0 

5 

7 

4 

8 

1 

7 

3 

68 

7 

3 
0 

2 

9»' 

8 

0 


4.  Supposing  a  wan  goes  a  journey,  and  on  the  Ist  day, 
1802.     May  14,  Pays  for  a  dinner     ----..- 

-  for  oats  for  his  horse      .... 

-  for  shoeing 

15,      -     for  supper  and  lodging   .... 

-  for  horse  keeping      -     -     .     -    . 

-  for  toll 

-  for  breakfast 

-  to  the  barber  for  dressing  .     -     - 

-  for  dinner  again  and  refreshment   - 

What  were  the  gentleman's  expeiwes  ? 


eo 

i 

e 

0 

0 

6 

0 

1 

2 

0 

2 

0 

0 

1 

10 

0 

1 

6 

0 

2 

0 

0 

1 

6 

0 

3 

5 

5.  Suppose  I  am  indebted  £. 

To  A.  Thirty-two  pounds  fourteen  shiUings  and  ten  pence. 

—  E.  Forty-one  pounds  six  shilHngs  and  eight  pence. 
-—  C.  Sevenly-five  pounds  eight  shillings. 

—  D.  Three  pounds  juidnine  pence., 


[hat  is  the  sum  I  owe  ? 


.ins. 


C.  A  man  purchases  cattle  ;  one  yoke  of  oxen  for  £14  116;  four  cows, 
fT.£l8  19  7;  and  other  stock  to  the  amount  of  £21  6;  what  was  the 
a  iiount  of  the  cattie  purchased  1  4ns,  £64  IGs.   Id, 


Sect.  I.  5. 


COMPOUND  ADDITION. 


87 


2.  OF  TROY  WEIGHT. 

By  Troy  Weight  are  weighed  gold,  silver,  jewels,  electuaries  and  liquors. 

TABLE. 

i  Pennyweight,    marked  pwt. 
make  one  <  Ounce,  or. 


24  grains  grs. 
fO  Pennyweights 
IS  Ounces 

1 

t,A 

th. 

70 

02.            pat. 
10              13 

grs 
4 

3 

9               7 

16 

f8 

0                0 

5 

7 

3                 6 

2 

^  Pound, 


lb 


EXAMPLES. 

Because  24  grains  make  a  penny- 
weight, you  carry  one  to  the  penny- 
weight column  for  every  24  in  the 
sum  of  tlie  column  of  grains  ;  be- 
cause 20  pennyweights  make  one 
ounce,  you  carry  for  20  in  penny- 
weights, and  because  12  ounces 
make  one  pound,  you  carry  for  12 
IQ  the  ounces.  This  is  called  car- 
rying according  to  the  value  of  the 
higher  place. 


2. 


lb. 

oz. 

pwt. 

lb.             oz. 

fXt. 

g^f. 

1  6  1 

7 

1  9 

7 

1  4 

2  3 

6 

5 

6 

2 

0 

C 

2  8 

0 

1  4 

1  1 

1  3 

5 

3 

7 

1  0 

1   2 

7 

JVotc. — The  fineness  of  gold  is  tried  by  fire,  and  is  reckoned  in  carats, 
by  which  is  understood  the  24th  part  of  any  quantity ;  if  it  lose  nothing  in 
the  trial,  it  is  said  to  be  24  carats  fine  ;  if  it  lose  2  carats,  it  is  then  22 
carats  fine,  which  is  the  standard  for  gold. 

Silver  which  abides  the  fire  without  loss  is  said  to  be  12  ounces  fine. — 
The  standard  for  silver  coin  is  11  oz.  2pwl3.  of  fine  silver,  and  18  pwti. 
jf  i-opper  molted  together. 


COMPOUND  ADDITION.  Sect.  I.  5 


3.  OF  AVOIRDUPOIS  WEIGHT. 

By  Avoirdupois  weight  are  weighed  all  things  of  a  coarse  and  drossy- 
nature,  as  tea,  sugar,  bread,  flour,  tallow,  hay,  leather,  and  all  itinds  of 
rnetals,  except  gold  and  silver. 

TABLE. 
16  Drams      dr.  \  /  Ounce,  marked  or. 

16  Ounces  /  V  Pound,  lb. 

28  Pounds  >  make  one  ^  Quarter  ofahund.  weight,  qr. 

4  Quarters  V  )  1 00 weight,or  1  Impounds, ca^ 

20  Hundred  weight      )  \  Ton,  T. 

EXAMPLES. 
1. 

T.  crt'f.  qr.  lb.  oz.  dr. 

186  3  2  25  II  8 

4  17  0  2  3  7  6 

9  8  3                   7  2  5 

2  3  1  16  5  11 


2. 


T. 

crut. 

qr. 

Ih. 

oz. 

dr. 

8  0  1 

3 

2 

2  5 

1  1 

8 

I 


7       19  3  14  5  6 

8  6  2  0  6  0         15 

3  7  1  0  6  4 


Note. — "  175  Troy  ounces  are  precisely  equal  to  192  Avoirdupois  Ounces, 
and  175  Troy  pounds  are  equal  to  144  Avoirdupois.  1.1b.  Troy=5760  grains, 
and  1  lb.  Avoirdupois=7000  grains." 


SscT.  I.  5. 


COMPOUND  ADDITION. 


99 


4. 

OF  TIME. 

- 

60  Seconds 
60  Minutes 
24  Hours 
7  Days 
4  Weeks 
13  Months, 

s. 
Id.fc  Ih.    , 

TABLE. 
•  make  one  • 

Minute,  marked 

Hour 

Day, 

Week, 

Month, 

♦  Julian  Year, 

m. 
/i. 

mo 
Y, 

EXAMPLES. 
1. 

F. 
1  6 

mo. 
1  0 

■w.                d. 

3             6 

h. 
23 

m. 
57 

s. 
43 

28 

7 

2             5 

1  6 

28 

32 

39 

6 

1              3 

1  7 

38 

1  1 

87 

4 

0              1 

1  4 

1  5 

1  7 

2. 


F. 

mo. 

TO. 

d. 

h. 

m. 

9. 

89 

1   1 

3 

6 

22 

45 

36 

36 

1  0 

2 

5 

6 

5  5 

44 

87 

'        2 

1 

'     0 

1  ] 

22 

33 

36 

4 

3 

3 

5 

8 

7 

The  number  of  days  in  each  Calendar  month  maybe  remembered  by  the 
following  verse  : 

Thirty  days  hath  September,  April,  June  and  November; 
February  twenty-eight  alone  ;  all  the  rest  have  thirty-one. 
*  "  The  civil  Solar  year  of  3G5  days  being  short  of  the  true  by  5h.  48m. 
67s.  occasioned  the  beginning  of  the  year  to  run  forward  through  the  sea- 
eon  nearly  one  day  in  four  years  ;  on  this  account  Julius  Cassar  ordainetl 
that  one  day  should  be  added  to  February  every  fourth  year,  by  causing 
the  24th  day  to  be  reckoned  twice  :  and  because  this  24lh  day  was  the 
sixth  (sextilis)  before  the  kalends  of  March,  there  were  in  this  year  two  of 
these  sexliles,  which  gave  the  name  of  Bissextile  to  this  year,  which  being 
thus  corrected,  was,  from  thence  called  the  Julian  year." 


40 


COMPOUND  ADDITION. 


Sect.  I.  B. 


GO  Seconds 
60  Minutes 
30  Degrees 
12  Signs,  or  360 
Degrees 


.'->.  OF  MOTION. 

TJIBLE. 

f  Prime  Minute,  marked  "  ' 

I    Degree,  ** 

'  make  one  <    Sign,  s. 

j  The  whole  great  circle  of  the 
L     Zodiac. 


EXAMPLES. 


2  5 

1  7 

6 

1  0 


1  7 

4  9 

3  5 

1  7 


1  8 

5  6 

2  4 
1  6 


8 
2  6 
1   8 

9 


6  5 

4  4 

3  6 

3  3 


4  4 

5  5 

1  2 

2  8 


6.  OF  CLOTH  IMEASURE. 


Inches,  one 

fifth    in. 

r  Nail,          marked  na. 

Nails,  or 

9  inches 

Quarter  of  a 

yard,  qr. 

Quarters 

of 

a  yard 

or  36  inches 

Yard, 

yd. 

Quarters 

of 

a  yard, 

or  27 

inches 

Ell  Flemish 

E.FI. 

Quarters 

of 

a  yard 

or  45  inches 

^make 

one  • 

Ell  English 

E.  E. 

Quarters  of 

a  yard, 

or  54  inches 

Ell  French 

E.  Fr. 

Quarters 

1  i 

nch  and  1  fiftl 

i,or  37 

inches 

and  one  fifth 

Ell  Scotch 

E.Sc. 

Quarters 

and  two  thirds 

,  Spanish  Var. 

1. 

EXAMPLES. 

2. 

Yds. 

qrs. 

n. 

E.  E.              qr. 

n. 

6  1  4 

3 

1 

3 

1  9                3 

2 

3  6 

1 

2 

6  6                 1 

3 

7 

0 

1 

7                 2 

2 

1   5 

3 

2 

1   8                 2 

0 

I  Sect.  I.  5. 


COMPOUND  ADDITION. 


41 


7.  OF  LONG  MEASURE. 

By  Long  Measure  are  measured  distances,  or  any  thing  where  leDgth  li 
considere  J  without  regard  to  breadth. 

TABLE. 


3  Barlej 

'  corns, 

fcar.    "] 

p   Inch, 

nuxrked 

in. 

12  Inches 

Foot, 

ft 

3  Feet 

Yard, 

yd. 

5\  Yards, 

or  ICh  feet 

Rod,  Perch  or  Pole, 

poU 

40  Poles 
8  Furlongs 

■  make  one  • 

Furlong, 
Mile, 

rmle 

(  Degree 
I           C-i 

of  a  great 

69i  Statute  miles, 

near/y. 

rcle. 

i  A  great 
-     t         the 

Circle  of 

SCO  Degree* 

Earth.   , 

EXAMPLES 
1. 

Des^. 

mi. 

y«r.             po/. 

ft- 

in. 

bar. 

1  6  8 

5  7 

7           2  6 

1   5 

1     1 

2 

1   2  4 

5  3 

6           1  8 

7 

6 

1 

7  9 

3  6 

1                7 

9 

I  0 

0 

4 

7 

3               0 

3 

2 

1 

2. 


jDc^. 

mi. 

fur. 

pol. 

y?. 

in. 

1   3 

5  6 

5 

1   3 

8 

1 

4  9 

1  8 

1 

2  7 

1  6 

2 

2  6  7 

1  2 

3 

I  6 

9 

0 

2  9 

8 

0 

5 

3 

1 

4S 


COMPOUND  ADDITION. 


Sect.  I.  6> 


8.  OF  LAND  OR  SQUARE  MEASURE. 

By  Square  Measure  are  measured  all  things  that  have  length  and  breadth. 

TABLE. 


144  Inches                     'j 

f   Square  Foot. 

>              9  Feet 

Yard. 

30]-  Yards,  or  ) 
2721  Feet          J 
40  Poles 

•  make  one  - 

Pole. 

Rood. 

4  Roods  1  f.O  rods  } 

or  4840  yards  I 

Acre. 

640  Acres                       J 

[     Mile. 

EXMIPLES. 

Acres.                          rood. 

pal.                          ft. 

iff. 

3  7  6                     3 

3  6                 9  3 

1  2  1 

5  6  8                     1 

2  7                 5  8 

7  6 

2  4  7                     2 

3  5                 6  1 

2  4 

9.  OF  SOLID  MEASURE. 

By  Solid  Measure  are  measured  all  things  that  have  length,  breadth  and 
thickness. 

TABLE. 


1728 

Inches 

1 

f   Foot. 

1-7 

Feet 

Yard. 

40 

OI 

Feet  of  round  timber  } 
'  50  feet  of  hewn  timber  \ 

>  make  one  ■ 

Ton  or 

load. 

128 

Solid 

feet,  i. 

e.   8  in) 

le 

ngth, 

4  in  breadth  and  ) 

Cord  of  wood. 

4 

in  bei 

.''^t 

)  J 

k. 

EXAMPLES. 

1. 

t 

2. 

Ton. 

/(' 

in.                  Cord. 

ft- 

in. 

e5 

3  7 

22  9            3  9 

1  1  8 

102  1 

?9 

2  6 

120  7               4 

56 

1  3  7 

Z6 

1  7 

54            18 

72 

6  59 

Z^7 



3  3 

G           2  9 

86 

1  24 

Sect.  I.  6. 


COMPOUND  ADDITION. 


10.  OF  WINE  MEASURE/ 

By  Wine  Measure  are  measured  Rura,  Brandy,  Perry,  Cider,  Mead, 
Vinegar  and  Oil. 

T.WLE. 

1  r 


2  Pints    pts. 

4  Quarts 
10  Gallons 
18  Gallons 
311  Gallons 
42  Gallons 
63  Gallons 

2  Hogsheads 

2  Pipes 


Hhd. 
3  9 

1  6 
3  5 

2  9 


marked 


}■  make  one  { 


Quart, 

Gallon, 

Anchor  of  Brandy, 

Runlet, 

Half  Hogshead. 

Tierce, 

Hogshead, 

Pipe  or  Butt, 

TUD. 


EXAMPLES. 
1. 


gal. 
5  2 

2  7 
1  2 

3  8 


qts. 

3 

1 

0 

2 


pts. 
1 
0 
1 
0 


r. 

hkd. 

ga/. 

^Is. 

pti 

8  6 

2 

5  3 

3 

1 

3  5 

1 

3  6 

1 

0 

1  7 

0 

2  9 

2 

1 

gal. 

anc: 

rwn. 
^hhd. 

tier. 

hhd. 

or  B. 

T. 


N.  B.    A  Pint  Wine  Measure  is  28|  cubic  inches. 


<» 


44 


COMPOUND  ADDITION. 


Sect.  I.  5".  »« 


11.  OF  ALE  OR  BEER  MEASURE. 

TABLE. 


N.  B.  A  Pint  Beer  Measure,  is  35i  cubic  inches. 


2  Pints                                "] 
4  Quarts 

Quart,             marked 
Gallon, 

gcK 

8  Gallons 

Firkin  of  Ale  in  London, 

A.fir. 

Z\  Gallons 

Firkin  of  Ale  or  Beer. 

9  Gallons 
2  FirkirH 

>  make  one  < 

Firkin  of  Beer  in  London,  B.^r. 
Kilderkin,                              kill. 

2  Kilderkins 

Barrel, 

bar. 

1i  Barrels,  or  54  gallons 

^  Barrels 

Hogshead  of  Beer, 
Puncheon, 

hhd. 
pun. 

3  Barrels,  or 2 hogsheads^ 

.  BuU, 

bun. 

EXAMPLES. 
1.                                                                    2. 
hhd.             gall.             qts.                             B.Jir.             gal. 
327             48             2                                  23                  6 

qfs. 
2 

2  8             5  13                                  4  5                 2 

3 

173             24              1                                   98                  T 

1 

2  7              16             0 

3  6                 8 

0 

iti^l' 


12.  OF  DRY  MEASURE. 

By  Dry  Measure  are  measured  all  Dry  Goods,  such  as  Corn,  Wheat, 

Seed,  Fruit,  Roots,  Salt,  Coal,  kc. 


T.WLE. 


2  Pints 
2  Quarts 
2  Pottles 
2  Gallons 
4  Pecks 
2  Bushels 
2  Strikes 
2  Cooms 
4  Quarters 
1J   Quarters 
n  Quarters 
2  Weys 


'  make  one  * 


Quart,             marked 

Pottle, 

Gallon, 

Peck, 

Bushel, 

qts. 
pot. 
gal. 

pk. 
hush. 

Strike, 

str. 

Coom, 

CO. 

QuaVter, 
Chaldron, 

ch. 

Chaldron  in  London. 

^'Vey, 

Last, 

StcT.  I-  6>.  COMPOUND  ADDITION.  45 


L, 

f:j.aji/pz,£5. 

]. 

2. 

=: 

— T 

6«s/». 

pk. 

qt. 

;jf. 

Ck. 

lush. 

pk. 

qts. 

2  7 

2 

6 

I 

3  7 

1  6 

2 

5 

1  8 

8 

7 

0 

2  6 

2  8 

3 

7 

2  0 

0 

I 

1 

1  8 

1  2 

1 

a 

1  0 

1 

3 

0 

1  7 

2  5 

3 

6 

\ 

_ 

N.  B.  A  gallon,  Dry  Measure,  contaios  268|  cubic  inches. 


JTie  follorvins^  are  denominations  of  things  cmiuted  by  the 

"  Tale." 

12  Particular  things  make  1  Dozen,  » 

I  „  12  Dozen 1  Gross, 

12  Gross  or  144  doz.       .   1  great  Gross. 

ALSO,  ., 

20  Particular  things  make  1  Score. 


Denominations  of  Measures  not  included,  in  the  Tables. 

6  Points  make  1  Line, 
12  Lines      .     .      Inch, 
4  Inches    .     .      Hand, 
I  3  Hands     .     .      Foot, 

6G  Feet,  or  4  Poles,  a  Gunter's  Chain, 
o  Miles      .     .      League. 

A  Hand  is  Msed  to  measure  Horses.     A  Fathom  to  measure  depths. 

A  League  iii  reckoning  distances  at  Sea. 

N".  B.  A  G.uintai  of  Fish  weighs  1  cwt.  Avoirdupois. 


46  COMPOUND  SUBTRACTION.  Sect.  I.  «. 

}  6.  COMPOUND  SUBTRACTION. 


COMPOUND  SUBTRACTION  teaches  to  find  the  difference  between 
any  two  sums  of  diverse  denominations.  I 

RULE  FOR  COMFOUND  SUBTRACTIOX. 

"  Place  those  numbers  under  each  other,  which  are  of  the  same  denom- 
"  ination,  the  less  being  below  the  greater ;  begin  with  the  least  deoomina- 
"  tion,  and  if  it  exceed  the  figure  over  it,  borrow  as  many  units  as  makd; 
"  one  of  the  next  greater ;  subtract  it  therefrom  ;  and  to  the  difference' 
•'  add  the  upper  figure,  remembering  always  to  add  one  to  the  next  supe- 
•'  rior  denomination  for  that  which  yoi^  borrowed. 

Proof. — In  the  same  manner  as  Simple  Subtraction.^      • 

'   ^'*   '  1.  .OF  MO^EY.      ^       ' 

1.  Supposing  a  man  to  h;ive  lent  j£i85  10s.  7c?.  ai^  to  have  received 
again  of  his  money,  £93  15s.  how  much  remains  due  ?     \ 


OPERAT10^. 


£. 

Lent          1  8  5 
Received      9  3 

1. 

s. 

1  0 
1  5 

d. 

7 
0 

£. 

From  3  1  0 
Take       8  5 

2. 
s. 

1  6 

Due              9  1 

1   6 

7 

^., 

Proof        18  5 

1  0 

7 

3. 

£.  s.  d. 

Lent  6  3  7  1  7  8         The  sum  of  the  sever::-!,  payments 

I must  first  be   added  toj^ether,  and 

/      1  6  3           2             6     the  amount  subtracted  ^:om  the  sum 
Received  V          7  8            4  lent, 
at  sundry  <J          19  15            11 
times.     /      1   3  9            6             8         4.    From  £39  7s.  6a. '1  7r.  take 
3  2  6   1            1             4     £7  13j.  I  Urf.  and  what  will  remain? 
"    Ans.  £31  J,^s.  63rf. 


I 


Received 
in  all 

Vet  d'.i'i 


Sect.  I.  6. 


COMPOUND  SUBTRACTION. 


47 


5.  A  certain  man  eold  a  lot  of  land  for  £735  lis.  Gd :  he  received  at 
one  time  £61  5s:  at  another  time,  £195  13s.  lid.  how  much  is  there  yet 
due?  Ans.£il8   Us.  Id. 


2.  OF  TROY  WEIGHT. 


2. 


^6. 
Frem  7  6 
Take         3 


Remains 
Proof. 


oz. 

8 

9 


prat. 
1  6 
1    7 


grs. 

1   3 

6 


lb. 

oz. 

pxct 

7 

3 

5 

2 

8 

9 

3.  OF  AVOIRDUPOIS  WEIGHT. 

1.  2. 


lb. 

oz. 

dr. 

r. 

cwt. 

or. 

Z6. 

or. 

dr 

9 

1  5 

6 

6 

1    1 

1 

1  4 

7. 

3 

6 

6 

7 

1 

5 

1 

1  6 

9 

8 

• 

r. 

3  9 

1  6 

mo. 
6 
9 

4.  OF  TIME. 
1. 

w.             d.             h. 
3               6              2  0 
1                2              18 

fn. 

4  4 

5  9 

6  5 
5  7 

48  COMPOUND  SUBTRACTION.  Seer.  I. 

5.  OP  MOTION. 

1.  2. 


1 


1  6 
8 

2  7 

3  4 

3  3 

2  S 

6 
3 

8 
9 

5  1 

6  7 

6.  0 

F  CLOT] 

a  MEASUI 

IE. 

1. 

2. 

Yd». 

qr. 

n. 

£.£. 

qr. 

«. 

2  7 

1 

2 

2  6 

2 

1 

1  6 

1 

3 

1    7 

3 

2 

, 

7.  OF  LONG  MEASURE. 

1. 

mi.          fur.            p.           yds           Jl. 
13              5           2  6             2               1 
15             2           2  7             1               2 

6  6 
1   7 

in 
8 
9 

fcar. 
1 

1 

8.  OF  LAND  OR  SQUARE  MEASURE. 


A. 
1  7 
1  6 

1. 
J?. 
1 

1 

pol. 
I   7 
1  6 

pol. 
1  8 
1  0 

2. 

1  6 
2  0  1 

tn. 
1   1 

I  3  0 

.....  f 

Sect.  I.  6.                COMPOUND  SUBTRACTION.  49 

OF  SOLID  MEASURE. 

Cords.            ft.  in. 

6  8               2  3  8  10 

6            12  7  15  2  9 


9. 

OF  SO 

1. 

7''ons. 
4  5 
1   9 

ft- 

2  9 

3  4 

in. 
1   8  6 
12  3  7 

10.  OF  WmE  MEASURE. 

1.  2. 

Hhd.  gal.  qts.  Tun.  Hhd.  gat. 

6  6  3   12 

17  3  3  3 


7  5 
2  4 

1 
1 

1   6 
4  3 

11.  OF  ALE  AND  BEER  MEASURE. 

1.  2. 

Hhd.            gal.  qts.  Butt.  hhd.  gal. 

8  9                19               2  6  3  1  16 

37               25               3  2-9  1  10 


1. 

6  1 

pk. 
1 

5 

1 

12.  OF  DRY  MEASURE. 

gls. 


0 


^ 

2. 

Chal. 

i«. 

pA-. 

1  7  1 

1   8 

1 

7  6 

.2  2 

2 

THE 


SCHOLAR  S  ARITHMETIC. 


OBSERVATIONS. 


The  Scholar  has  now  surveyed  the  ground  work  of 
Arithmetic.  It  has  before  been  intimated  that  the  only 
way  in  which  numbers  can  be  affected,  is  by  the  operations 
of  Addition,  Subtraction,  Multiplication  and  Division. 
These  rules  have  now  been  taught  him,  and  the  exercises  in 
a  supplement  to  each,  suggest  their  use  and  application  to 
the  purposes  and  concerns  of  life.  Further,  the  thing  need- 
ful, and  that  which  distinguishes  the  Arithmetician,  is  to 
know  hoAv  to  proceed  by  application  of  these  four  rules  to 
the  solution  of  any  arithmetical  question.  To  afford  the 
scholar  this  knowledge  is  the  object  of  all  succeeding  rules. 


SECTION  II. 

F.l'LES     ESSENTIALLY    NliCESSARV    FOR    EVERY    PERSOV    TO    FIT    AND    ftUALIFY 
THEM    FOP.    THE    TRANSACTION    OF    BUSINESS.  '^ 

These  are  ten :  Reduction,  F^-actions*  Federal  Money,  Exchange,  Interest, 
Compound  Multiplication,  Compound  Divisio7i,  Single  Rule  of  Three,  Double 
Rule  of  Three,  and  Practice. 

A  thorough  knowledge  of  these  rules  is  sufficient  for  every  ordinary  oc-. 

currence  in  life.     Short  of  this  a  person  in  any  kind  of  business,  will  be 

liable  to  repeated  embarrassments.     It  is  the  extreme  usefulness  of  these 

rules  which  commends  them  to  the  attention  of  every  Scholar. 

*  FnAcritms  are  takt-n  up  here  no  further  than  is  necessary  to  shcAV  their  signification, 
aii'l  to  ilhistratp  the  principles  of  IV.dkral  Monkv. 
i 


Sect.  IL  1.  REDUCTION.  51 

i  1.  REDUCTION. 


"  REDUCTION  teaches  to  bring  or  exchange  numbers  of  one  denom- 
"  ination  to  others  of  different  denominations,  retaining  the  same  value." 

IT   IS   OF   TWO    KINDS. 

1.  Wfien  high  denominations  are  to  be  hroxight  into  lower,  as  pounds  into 
shillings,  pence  and  farthings  ;  it  is  then  called  redvction  descendinq,  and 
is  performed  by  Muliiplicalion, 

2.  I'ilieii  lower  denominations  are  to  be  brought  into  higher,  as  farthings  into 
pence,  or  into  pence,  shillings  and  pounds  ;  it  is  then  called  reduction  as- 
cending, and  is  performed  by  Division. 

REDUCTION  DESCENDING. 

RULE. 

Multiply  the  highest  denomination  by  that  number  which  it  takes  of  the 
next  less  to  make  one  of  that  greater ;  so  continue  to  do  till  you  have 
brought  it  as  low  as  your  question  requires. 

Proof — "  Change  the  order  of  the  question,  and  divide  your  last  pro- 
duct by  the  last  multiplier,  and  so  on." 

EXAMPLES. 
1.  In  £17  13s.  6 J.  3qrs.  how  many  farthings  ? 

OPERATION. 

£.      s.      d.      grs.  In  this  example,  the  highest  denom- 

1  7     13      6         3  ination  is  pounds,    the  next    less,    is 

2  0  Shillings  in  a  pound,  shillings,    and    because    20    shillings 
make  one  pound Aherefore,  I  multiply 


6  3  S}iillingsin£\l  13s.  £17  by  20,  increasing  the  product  by 

1   2  Pence  in  a  Shilling,     the    addition    of  the    given  shillings, 

(13)  which  it  must  be  remembered, 


4  2  4  2  Pence  in£,n  13s.  Gd.  must  always  be  done    in  like  cases, 
4  Farthings  in  a  penny,  then  because   12  pence  make  one  shil- 
ling, I  multiply  the  shillings  (353)  by 


.^.16971  Farthings.  12,  adding  in  the  given  pence  (6rf.) 

lastly,  because  4  fahhings  make  one  penny,  I  multiply  the  pence  (4242) 
j^Y  4.  and  add  in  the  given  farthings  {3qrs.)  1  Ihrn  find  that  in  £17  13s.  Gd. 
oqrs.  there  are  16971  farthings. 

PROOF. 

3qrs.  To  prove  the  above  question,  change  the 

order  of  it,  and  it  will  stand  thus  :  in  16971 
Gd.  farthings,  how  many  pounds  ? 

<^ivide  the  last  product  by  the  last  multiplier, 

13s.  the  remainder  will  be  firthings.     Proceed  in 

this  way  till  all  the  steps  of  the  operation  hare 

£1   7  been  retraced  back  ;  the  last  quotient  with  the 

remainders  will  be  proof  of  the  accuracy  ol 

the  operation  if  they  agree  v.ith  tlie  sum  given  ia  the  question. 


4)   1   6 

9 

7   1 

12)  4 

2 

4  2 

2|0) 

3 

513 

REDUCTION. 


Sect.  II.  L 


2.  In  jC?  14*.  6d.  Iqr.  how  many 


farthiogi 


Ant.  14nqrs. 


3.  In  £l  6s.  4d.  how  many  pence"? 
Jins.  llOQd, 


4.  In  29  guineas,  at  SJ8s.  each,       5.  In  £173  15j.  howmany  six-]j*en 
how  many  farthings?  Ans.  3891i5  qrs.     ces  2  Ans.  6d50. 


6.   In   12  crowns  at  Gs.  Id.  how 
i&any  peiace  and  farthings  ? 

Ans.  948c/.  tild2qrs. 


7.  In  671  eagles,  at  10  dolls,  each, 
how  many  shillings,  three-pencep, 
pence,  and  farthings  ?  Ans.  40260s. 
161040  thrce-pences,  483120  pence ^ 
and  1932480yrs. 


«> 


Sect.  II.  I.  REDUCTION.  '  5^ 

REDUCTION  ASCENDING. 

RULE. 
Divide  the  lowest  denomination  given  by  that  number  wb;eh  it  takes  of 

the  same  to  make  one  of  the  next  higher,  and  so  continue  to  do  till  you  have 
brought  it  into  tJie  denomination  which  your  question  requires. 

EXJiMPLES. 
1.  In  16971  farthings  how  many  pounds  ? 

OPERATION. 

Farthings  in  a  penny     4)16971  Zqrs. 


Pence  in  a  shilling        12)4242  6(£.  Reduction  descending  5J»d  ascend- 

ing  reciprocally  prove  each  otherc 

Shillings  in  a  pound       2j0)35|3  \3s. 

£17 
Ani.£\l  13s.  Gd.  Zqrs. 


2.    In    1765  pence,    how   many         3.  In  38976  ftrthings  how  many 
pounds  ?  Jins,  £7  7s.   \d.         guineas  ?  Jins.  29 


4.  In  6950  sixpences,  how  many         5.  In  3792  farthings,  how  many 

pounds?  .4ns.  £173  15.?.        crowns?  Mt.  U. 


64 


REDUCTION.  Sect.  II.  i. 


G.  In  ISneO  farthing,  how  many         7.     In    6952   three-pences,   how 
pence,  thrce-pences,  six-pences  and     many  pistoles  at  22x.         Ms.  79. 
dollars  ? 
Ans.  12240  pence, 40BO  three-petices, 
2040  sii-pences,  170  dollars. 


REDUCTION  ASCENDING  Sf  DESCENDING. 

1.  MONEY. 
J.  In  57  moidores,  at  365.  each,        In  this  question  the  first  step  will 
how  many  dollars  ?  be  to  bring  the  moidores  into  shillings: 

Ans.  ^342.        lastly  bring  the  shiUings  into  dollars. 

2.  In  75  pistoles  how  many  pounds  ? 
Ans.  £82   10s. 


3.  In  jG73  how  many  guineas  ?  4.  In   £,C'3  and  o  guineas,    how 

Aas.  52  guineas,  4s.  many  dollars  ?     Ans.  ^233  2*. 


SrcT.  II.  1.  REDUCTION.  65 

"  When  it  is  required  to  know  how  many  sorts  of  coin  of  di^erent  values 
and  of  equal  number  are  contained  in  any  number  of  another  kind ;  reduce 
the  several  sorts  of  coin  into  the  lowest  denomination  mentioned,  and  add 
them  together  for  a  divisor  ;  then  reduce  the  money  given  into  the  same 
denomination  for  a  dividend,  and  the  quotient  arising  from  the  division  will 
be  the  number  required." 

Note.     Observe  the  same  direction  in  weights  and  measures. 

1.  In  54  guineas,  how  many  pounds,  dollars  and  shillings  of  eac^  an 
equal  number  ? 

OPERATION. 

£1  is  20  shillings  54  guineas 

1  dollar  is    6  shillings  28  shillings  is  a  guinea 

1  shining  is  1  shilHug  

_  432 

Divisor  27  shillings  108 

Dividend  1512  shillings. 

27)1512(56  of  each  ;  that  is,  54  guineas  include  the  value  of  one  pound, 
135         one  dollar,  and  one  shilling  56  times. 

162 
162 

000 

2.  In  1 72  moidores  how  many  eagles,  dollars  and  nine-pences,  of  each  the 
like  number?  Jlns.  92  of  each,  and  68  nine-pences  over. 


3.  In  237  guineas  how  many  moidores,  pistoles,  pounds,  and  dollars,  ot 
each  the  like  number  ?  -Ins,  79  of  each. 


66  REDUCTION.  Sect.  II.  I 

TROY  IVEIGHT. 
•,    1.  In  4/6.  boz.  and  IGpvts.  how  many  grains  ? 

OPERATION. 

Ih.       oz.       pK'U 

4  6  16 

12  oz.  in  a  pound. 

63  ounces. 

20  pwts.  in  an  ounce. 

1076  penny  wei<!;hts. 
24  grains  in  1  pwt. 


4304 
2152 


froo/ 24)25824  grains,  the  Ans. 
20)1076     16  pwts. 

12)53     6oz. 


4lb. 


2.  In  lOlh.  of  silver,  how  many  spoons,  each  weighing  Boz.  10  pwts.  1 

Ans.  21  spoons y  and  QOpxmis.  over. 


S.  In  282240  grains  of  silver,  how  many  pounds  ?  Ans.  49. 

pnooF. 


Sect.  II.  1.  REDUCTION.  67 

4.  In  45681  grains  of  silver,  how  many  porunds  ? 

Jnswer  lib.  11  or-.  3p^ts,_  Bgrs. 


5.  la  4560  grains  of  silver,  how  many  tea  spoons,  each  one  ounce  1 

Ans^.  9^  tea  spoons. 


PROOF, 


Cwt. 

1.  In  67 

4 

269 
28 

3. 
qr. 
I 

AVOIRDUPO 
Ih,         oz. 
13          11, 

IS  WEIGHT. 
how  many  drams  ? 

PROOF. 

16)1931696 

2165 
538 

7545 
16 

16)120731      l\oz 

28)7545     13/S 

4)269      \qr. 
67    Cvct 

45^81 
7545 

120731 
16 

724386 
120731 

1931696 

H 


"  REDUCTION.  Sect.  II.  1^ 

«.  In  UOiSoz.  how  many  hundred  weight  ?  "  An$.  IC  Sqrs.  10/6. , 

f 


3.  In  470  boxes  of  Sugar,  each  26^6.  how  many  Cwt.  ? 

Jins.  109C.  Oqrs.  12/6. 


4.  in  nCwi.  Iqy.  &lb.  of  Sugar,  how  many  parcels,  each  17/6.  ? 

Ans.  114  parcels. 


^ 


Sect.  II.  1.  REDUCTION.  •     59 

4.  TIME. 
1.  In  121812  seconds  how  many  hours  ? 

OPERATION.  PROOF. 

6|0)12181|2     12  sec.  H.     m.     s. 


33     50     12 


6,0)203|0       50m.  60 

2030 

.9ns.  33h.  60m.   12s.  60 

121812 

2.  Supposing  a  man  to  be  21  years  old,  how  many  seconds  has  he  lived, 
allowing  366  days,  6  hours  to  a  year  ?  Ans.  662709600  seconds. 


3.  How  many  minutes  from  the  commencement  of  the  war  between 
America  and  England,  April  19,  1775,  to  the  settlement  of  a  gone'ral 
peace,  which  took  place,  January  20,  1783  ?  Ans.  4079160  minutes. 


60  REDUCTION. 

4.  In  413280  minutea  how  many  weeks  ? 


Sect.  II.  l^ 
^ns.  41  Tecakt. 


5.  LONG  MEASURE. 
6.  Reduce  IQ>  miles  to  barley  cornsi 


OrEIlATION. 

16  Mies. 


128  Furlongs. 
40 

5120  Rods. 
6i* 


25600 

2660 


28160  Yards. 
3 


34480  Feci. 
12 


1013760  Inches. 
3 


pnoor. 
3)3041280 
12)1013760 

3)84480 

111)28160 


2560 


4{0)512j0 


8)1£8 


16  MiUs. 


t  Divide  by  11  for  5^  and  multiply 
tlie  quotient  by  2.  TJie  reason  is  be- 
cause 5i  reduced  to  half  yards  is  1 1-. 


.^risa-er,  3041280  bar.  corns. 
*  To  multiply  by  one  half  (^)  it  is  only  to  take  half  the  multiplicand. 
2.  In  47520  feethow  many  leagues  ?  Ans.  3  leagues. 


Sect.  II.  1.  REDUCTION.  61 

3.  How  many  times  does  the  wheel  which  is  18  feet  6  inches  in  circum- 
ference, turn  round  in  the  distance  of  150  miles  ? 

Jins.  42810  times,  and  |80  inches  over. 


4.  How  many  barley  corns  will  readli  round  the  Globe,  it  being  360 
degrees?  ^ms.  47558016Q0. 


6t 


REDUCTION. 


Sect.  II.  1. 


6.  LAND  OR  sqUARE  MEASURE. 

1.  In  13  acres,  2  roods,  how  many  poles  ? 

OPERATION.  PROOF. 

Ac.         R.  4)0)2 16|0 

13  2  

4 

64 
40 

Ans.  2160  Poles. 

2.  In  2862  rode  how  many  acres  ? 


4)54 

13Ac.  2R. 

Ans.  11  A.  3R.  12P 


7.  SOLID  MEASURE. 
1.  Ih  1296000  solid  inches  how  many  tons  of  hewn  timber  ? 

OPERATION. 

6|0 
1728)1296000(75|0 
12096     


8640 
8640 

00 


15  Tons,  the  Answer. 


PROOF. 

15 

50 

760 
1728 

6000 
1500 
6250 
760 

1296000  Inches. 


Sect.  II.  1.  REDUCTION.  63 

2.  In  5529600  solid  inches,  how  many  cords  of  wood  ?  Ans.  25. 


3.  How  many  solid  inches  in  a  cotd  ?  -'ins.  22 11 84. 


8.  DRY  MEASURE. 
1.  In  75  bushels  of  com  how  many  pints  ? 


OPERATION. 
75 

4 

300 
8 

2400 
2 

PROOF. 

2)4800 

8)2400 

4)300 

75  Bushels. 

Ans.  4800  pis. 
2.  In  9376  quarts  how  many  buehels  ? 

Ans.  293 

It  would  be  needless  to  give  examples  of  Reduction  in  all  the  weights 
and  measures.  The  understanding  which  the  attentive  Scholar  must 
ah-eady  have  acquired  of  this  rule,  by  the  help  of  the  tables,  will  ever  be 
si^hcient  for  his  purpose. 


64  SUPPLEMENT  TO  REDUCTION  Sect.  II.  1  Hi 

SUPPLEMENT  TO  BUDUCTION. 


QUESTIONS. 
1 .  What  is  Reduction  ? 
'2.  Of  how  many  kiads  is  Reduction  ?    What  are  they  called  ?    Wherein 

do  these  kinds  differ  one  from  the  other  ?  Which  of  the  fundamental 

rules  are  employed  in  their  operation  ? 

3.  How  is  Reduction  Descending  performed  ? 

4.  l!ow  is  P».cduction  Ascending  performed  ? 

b.  When  it  is  required  to  know  how  many  sorts  of  coin,  weights  or 
measures  of  different  values,  of  each  an  equal  number,  are  contained 
in  an}'  other  number  of  another  kind,  what  is  the  method  of  pro- 
cedure ? 

EXERCISES. 

1.    lu   36   guineas,     how    many        2.  How  many  rings,  each  weigh- 
Ci'OWns  ?  ing  bp'wts.   Igrs.   may  be    made   of 

Jlns.  163  crotf?j«t^  9  J.  over.  oil.  ooz.  IGvz^ts.  2^rs.  of  gold? 

-iv.  .'ins.  loo. 


i 


SrcT.  II.   I.  SUPPLEMENT  TO  REDUCTION.  '         65 

3.  How  many  steps  of  2  feet  5  inches  each,  will  it  require  a  man  to  take, 
travelling  from  Leominster  to  Btston,  it  being  43  miles  ? 

Ans.  93947. 


4.  Let  70  doll^rg  be  distributed 
among  three  men  in  such  manner 
that  as  often  as  the  first  has  os.  the 
second  shall  have  75.  and  the  third  9*. 
What  will  each  one  receive  ? 

Jlns.  first  gl6  4s.  second"  ^23  2j, 
fjiird  $30. 


5.  How  many  squjare  feet  in  a  square 
mile  ?  Ms.  27878400. 


66  SUPPLEMENT  TO  REDOCTION.  Sect.  II.  1. 

6.  If  a  vintner  be  desirous  to  draw  off  a  pipe  of  Canary  into  bottles,  con? 
laining  pints,  quarts,  and  2  quarts,  of  each  au  equal  number,  how  many 
mast  he  have?    v  -  Ans.  144  of  each. 


7.  There  are  three  fields,  one  contains  7  acres,  another  10  acres,  ancl 
the  other  12  acres  and  1  rood:  how  many  shares  of  76  perches  each,  are 
contained  in  the  whole  2  Jins.  61  shares  and  44  perches  ovev. 


Sect.  II.  1.  SUPPLEMENT  TO  REDUCTION.  67 

8.  There  a/e  106/6.  of  silver,  the  property  of  3  men;  of  which  A  re- 
ceives 17Z6.  lOoz.  I9pwts.  19grs.  of  what  remains,  B  shares  loz.  Igrs.  so 
Qflen  as  C  shares  ISpruts.     What  are  the  shares  of  B  and  C  ? 

Ans.  B's  share  53/6.  Boz.  Spwts.  5grs.     Cs  share  Hlb.  4oz,  ISpaft. 


FRACTIOXS.  Sect.  LI.  i\ 

^  2.  FRACTIONS. 

■WHEN  tlie  thing:  or  things  signUietl  by  figures  are  whole  ones,  then  (he 
fii^'jrcs  which  signify  them  are  called  integers  or  tc7(o/e  numbers.  But 
wliun  only  seme  pans  of  a  thing  are  .sigiiified  by  ligures,  as  t%!;o  thirds  of 
-.iDY  liiiiK.r,  ^i-'c  iixths,  seven  fcnths,  i5'f.  then  the  figures  which  signify  these 
I'Ciris  of  a  thing  being  the  expression  ai'  some  quantity  less  than  one,  are 
ctilled  FRACTior.s.         • 

Fractions  are  of  two  kind;*,  Fuigar  and  Decimal;  they  are  distinguish- 
ed bv  the  manner  of  representing  them;  they  also  differ  in  their  modes  of 
operation. 

VULGAR  FRACTIONS. 

To  understand  Vulgar  Tractions,  the  learner  must  suppose  an  integer 
(or  tiic  number  1)  divided  into  a  number  of  equal  pa^-ts  ;  then  any  number 
t'f  these  parts  being  taken  >vould  make  a  fraction-,.',  which  would  l>e  re- 
pre-cuted  by  two  luimbers  placed  one  directly  over  the  other  with  a  thort 
lino  between  them  thus,  -|  tuo  thirds,  ^-  three  Hflhs,  {-  seven  eic;hths,  ^'c. 

Each  of  thesfl  figures  have  a  difierent  name  aiid  a  dill'erent  signification. 
The  figure  below  liie  line  is  called  the  denominator,  and  shews  into  how 
m.'.iiv  parts  an  integer,  or  one  individual  of  any  thing  is  divided — tht:  figurti 
••ibove  the  line  is  calh^d  the  numerator,  and  shews  how  many  of  those  parta 
are  signified  by  the  fraction. 

For  iilustiation,  suppose  a  silver  plate  to  be  divided  into  r\>ne  equal  parts. 
Xo'Stone  or  more  of  these  parts  make  a  fraction  which  will  be  represented 
bv  liie  figure  9  for  a  denominator  placed  underneath  a  short  line  shewing  tho 
plate  to  be  divided  into  nine  equal  parts ;  and  supposiiig  tn-o  of  those  part:* 
(0  be  lakcu  for  the  fraction,  then  the  figure  2  must  be  placed  directly  abovi*. 
the  y  iind  o\'er  the  line  (f)  for  a  numerator,  showing  that  two  of  those  parts 
are  signified  by  the  fraction,  or  iv^o  ninths  of  the  plate.  Now  let  o  parts  of 
this  plate,  which  is  divided  into  9  parts  be  given  to  Joh.n,  his  fraction  would 
be  ^  free  ninths;  let  3  other  parts  be  given  to  Harry,  his  fraction  would  be 
ij  three  ?iinths ;  there  would  then  be  one  part  of  the  plate  remaining  still 
(5  and  3  are  8)  and  this  fraction  would  be  expressed  thus  {;  one  ninth. 

In  this  way  all  vulgar  fractions  are  written  ;  the  denominator  or  number 
below  the  liiie,  L-hewing  into  how  many  parts  an}'  thing  is  divided,  and  the 
nunjcrator,  or  number  above  the  line,  shewing  how  many  of  those  parts 
itre  taken  or  signified  by  the  fraction. 

To  ascertain  whether  the  learner  understands  what  has  novv  been  taught 
hill)  of  iVactions,  let  us  again  suppose  a  dollar  to  be  cut  into  13  equal 
parts  ; — let  2  of  these  parts  be  given  to  A  ;  4  to  B  ;  and  7  to  C. 

i  A's  fraction  — 
Required  of  tiie  learner  tliat  he  should  write    <f   IVs  fraction  — 

^   C's  fraction  — 
It  is  from  divi^-.ion  only  lljut  fractions  arise   in  Arithmetical  operations  : 
the  )C;iiainder  after  division  is  a  portion  of  the  Dividend  undivided  ;  and  is 
al»;'v   t!ie  numerator  to  a  fraction  of  which  the  Divisor  is  the  Denominator. 
The  ';i.')tient  is  so  many  integers. 

The  Aritiimetic  of  Vulgar  Fractions  is  tedious  and  even  intricate  to  be- 
ginners. Besides  they  are  not  of  necessaiy  use.  We  shall  not  thereforo 
enter  into  any  further  consideration  of  them  here.  This  difficulty  arises 
(■!ii;:flY  from  the  varietv  of  deiiuminators  :  for  when  numbers  arc  divided 


5kct.  II.  2.  DECIMAL  FRACTIONS.  Q-' 

into  difiercnt  lands,  or  parts,  they  cagnol  be  easily  compared.     This  con- 
sitlei'ation  gave  rise  to  tlic  invention  of 

DECIMAL  FRACTIONS, 

Docirp.nl  Fraciioii>.  arc  a!.-o  expressions  of  parts  of  an  integer;  or  are  v.: 
laliie  soiacti)injj  k-ss  than  one  of  any  tliins",  %vhalover  it  may  be  v.hich  i^ 
signilled  by  ttion!, 

in  decini.ils  an  inicc;er,  or  tlie  number  otic,  as  1  foot,  !  dollar,  1  year,  &ir, 
is  conceiv(ul  to  be  divided  iuto^!.'/;  equal  parts,  (ui  vuli^arfraciions,  an  inte- 
ger may  he  divided  into  any  number  of  parts)  and  each  of  tliese  parts  is  sub- 
divided into  ten  lesser  parts,  and  so  on.  lo  this  way  the  denoirnnator  to  a 
decia^.al  fraction  in  all  r.-^ges,  will  be  either  10,  100,  1000,  or  unity  (1) 
H'ilh  a  niunbcr  of  cyphers  annexed  ;  and  this  nunibcc  of  cyphers  will  aluaya 
be  equal  to  the  nnmbcr  o(  places  in  the  numerator.  Thus,  /^  fj^  lY,/^^ 
zsc  Derimnl  Fractions,  of  whicjj  the  cyphers  in  the  denoniicator  of  each  arii 
equid  to  the  number  of  places  in  its  own  numerator. 

"  As  the  deno:r.inator  of  a  decimal  fraction  is  always  10,  100,  1000,  &>'. 
'  the  denominators  ncediiot  be  expressed  ;  for  the  numerator  only  maybe 
'•  made  to  express  the  true  value  ;  for  this  purpose  it  is  only  ro;p,ireu  to 
'"  write  the  numerator  with  a  point  (,)  before  it,  called  a  separatrix,  at  the 
"  kit  hand,  to  distins^uish  it  from  a  whole  number  ;  thus,  ."j  is  written  ,6  ; 

Vv'iicn  inteirers  and  deciniais  are  expressed  together  in  the  same  sui.:. 
iliat  suih  is  called  a  ??j/,rf(/ number  ;  thus,  25,^3  is  a  mixed  number ;  2n. 
or  all  the  tigures  to  the  left  hand  of  the  separalrix  being  integers,  and  ,t  ._> 
or  all  the  fii^ires  to  the  right  hand  of  the  same  point  being  decimals. 

The  first  tigure  on  the  right   hand  of  the   decimal  point  signifies  tenlh 
parts  ;  the  next,  hundredth  parts  ;  the  next,  thousandth  parts,  and  so  on. 
,7  seven  signifies  seven  tenth  parts. 
,(>7     —      seven  hundredth  parts.- 

,"7     —     two  tenth  parts  and  seven  hundredth  parts  ;  or  twenty  seven 
hundredths. 
,.3o7     —     three  tentli  parts,  five   hundredth  parts,  and  seven  thoui- 
andth  parts  ;  or  357  thousandths. 
5,7     —      five,  and  seven  tenth  parts. 
5,007     —      five  and  seven  thousandths, 
i    The  value  of  each  figure  from  unity,  and  the  decrease  of  decimals  Iq- 
ward  the  right  hand  may  be  seen  in  the  following 

TABLE. 

'/)    'A    -r. 

^   G   S   "^   'c  <"  'o 

«   ^_  ,-.  '_  t»  T*  r* 


r«      t- 


c    =: 


rr     ^     ^     j;      --     vj     w^  r^     K*     v-     '*-      ^J 

J—  —  ■--.ococc.t;:;;cooo~  —  — 

OX     ox  xo     xo 

.9  87G5  1321,23450739 

Cyphers  placed  to  the  right  band  of  decimals  do  not  alter  their  value,-' 
Placed  at  the  left  band  Ihey  dimini-h  thoir  vabie  in  a  tenfold  proportion. 


70 


DECIMAL  FRACTIONS. 
ADDITION  OF  DECIMALS. 


Sect.  II.  2 


-  >  RULE. 

"  I.  Place  the  aumbers  whether  mixed  or  pure  decimals>  under  each 
"  other  according  to  the  value  of  their  places." 

"  2.  Find  theiy  sum  as  in  whole  numbers,  and  point  off  so  many  places 
"  for  decimals  as  are  equal  to  the  greatest  number  of  decimal  places  in  any 
"  of  the  given  numbers." 

EXAMPLES. 

1.  What  is  the  amount  of  73,612  guineas,  436  guineas,  3,27  guineas, 
,8632  of  a  guinea,  and  100,19  guineas  when  added  together. 

OPERATION.  The  decimals  are  arranged  from 

73,612  the   separatrix    towards    the    right 

436,  hand,  and   the  whole   numbers  from 

3,27  the  same  point  towards  the  left  hand. 

,8632  The    greatest   number   of   dcciro.al 

100,19  places  in  any  of  the  numbers  is  four, 

consequently  four  liguros  in  the  pro- 
duct must  be  pointed  ofl'for  decimals. 


Ans.  613,9352  guineas. 

2. 

345,601 
,3724 
63,1 
672,313 
7,5462 


3.  Required  the  sum  of  37,82 1-1- 
646,36-f8,4-f37,325. 

Ans.  620,896. 


4.  What  is  the  sum  of  three  hun- 
dred twenty-nine  and  seven  tenths  ; 
tliiirty-seven  and  one  hundred  and 
sixty-two  thousandths  ;  and  sixteen 
hundredths,  when  added  together  ? 
Ans.  367,022, 


&.  Add  six  hundred  and  five  thou- 
sandths, and  four  thousandth  and 
thre£  hundredths  ? 

Sum  4600,035. 


Note. — When  the  numerator  has  not  so  many  places  as  the  denominator 
has  cyphers,  prefix  so  many  cyphers  at  the  left  hand  as  will  make  up  the 
defect  i  so  -^f^-^  is  written  thus,  jOOj,  kc. 


Sect.  II.  2:  DECIMAL  FRACTIONS.  71 

SUBTRACTION  OF  DECIMALS. 
RULE. 
"  Place  the  numbers  according- to  their  value  ;  then  subtract  as  io  whele 
numbers,  and  point  off  the  decimals  as  in  Addition." 

EXAMPLES. 
1.  From  716,325  take  81,6i?01.  2.  From  119,1384  take  95,91. 

OPERATION.  Kern;  23,?284. 

From  71(3,325 
Take      81,6201 


634,7049 


3.  What  is  the  difference  between  4.  From  67,  take  ^92. 

207  and  3,115  ?         Arts.  283,885.  Ron.  66,  08. 


All  the  operations  in  Decimal  Fractions  are  extremely  easy ;  the  onljr 
liability  to  error  will  be  in  placing  the  numbers  and  pointing  otT  the  deci- 
ipalfs ;  and  here  care  will  always  be  security  against  mistakes. 

MULTIPLICATION  OF  DECIMALS. 

RULE. 

"  1.  Whether  they  are  mixed  numbers  or  pure  decimals,  place  the  fac- 
tors, and  multiply  them  as  in  whole  numbers." 

"  2.  Point  off  so  many  figures  from  the  prodact  as  there  are  decimal 
places  in  both  the  factors  ;  and  if  there  be  not  so  many  decimal  places 
in  the  product,  supply  the  defect  by  prefixing  cyphers." 

EXAMPLES. 

1.  Multiply  ,0261  by  ,Q035.                In  tl?is  example,  the  decimals  in  the 

OPERATION.  two  factors  taken  together  are  eight; 

,0261  the  product  falls  short  of  this  number 

,0035  by   four  figures,   consequently,  four 

■  cyphers  are  pretixed  to  the  left  ban(!l 

1305  of  Uie  product. 
783 


,00009135  Product. 


72 


DECIMAL  FRACTIONS. 


Sect.  II.  £. 


J.  Multiply  31,72  by  G5,3. 
Product,  2071,316. 

OPERATION. 

3  1,72 
6    5,  3 


Multiply  25,238  by  12,17. 
Product,  307,14646. 


Multiply  ,62  by  ,04. 
Product,  ,0248. 


Multiply  17,6  by  ,75. 
Product,  13,2. 


DIVISION  OF  DECIMALS. 
RULE. 

"  1.  The  places  of  decimal  parts  in  tl;e  divisor  and  quotient  counted 
together  must  be  always  equal  to  those  in  the  dividend,  therefore  divide  as 
in  whole  numbers,  and  from  the  right  hand  of  the  quotient,  point  ofi'  ?o 
many  places  for  decimals,  as  the  decimal  places  in  the  dividend  exceed 
those  in  the  divisor. 

"  2.  If  the  places  of  the  quotient  be  not  so  many  as  the  rule  requires, 
supply  the  defect  by  prefixing  cyphers  to  the  left  Land. 

"  3.  If  at  any  time  there  be  a  remainder,  or  the  decimal  places  in  the 
divisor  be  more  than  those  in  the  dividend,  cyphers  may  be  annexed  to  the 
dividend  or  to  the  remainder,  and  the  quotient  carried  on  to  any  degree  of 
exactness." 


Divide  2,735  by  51, S 

OPERATIOX. 

51,2)2,735(,0534  + 
2,560 


1750 
1536 


2140 
2048 

92 


EXAMPLES. 


In  this  example  there  zrejive  decimals 
in  the  dividend  (counting  the  two  cy- 
phers which  were  added  to  the  remain- 
der of  the  dividend  after  the  first  division) 
that  the  decimals  in  the  divisor  and  quo- 
tient counted  together  may  equal  that 
number,  a  cypher  is  prefixed  to  the  left 
band  of  the  quotient. 


Si;cT.  II.  2.  DECIMAL  FRACTIONS.  73 

In  the  division  of  decimals  it  is  proper  to  add  cyphers  so  long  as  there 
continues  to  be  a  remainder,  this  however  is  not  practised,  nor  is  it  neces- 
sary ;  four  or  live  decimals  being  sufficiently  accurate  for  most  calculations. 

2.  Divide  315G,293  by  25,17. 
Quotient,  125.34- 

NoTE.  Tlie  separatiix  is  omitted  in 
the  ans\rrrs  to  the  exiunples  on  this 
pagn  to  exercise  the  scholar  in  placing 
it  according  to  rule ;  to  this  the  In- 
structor should  be  jwirticularly  atteu- 
live. 


3.  Diride  5737  by  13,8. 
Quotient,  431353-f  _ 


Divide  173948  by  ,375. 
Quotient,  463C61-f 


5.  Divide  2  by  63,1 
Quotient,  037+ 


G. 'Divide  ,012  by  ,005. 
Quotient,  24. 


74  DECIMAL  FRACTIONS.  Sect.  II.  2. 

REDUCTIO.^  OF  DECIMALS. 

CASE  1. 

TO  REDUCE  VULGAR  FRACTIONS  TO  DECIMALS. 
RULE. 
Annex  a  cypher  to  the  numerator  and  divide  it  by  the  denominator,  an- 
nexing a  cypher  contiaually  to  the  remainder-     The  quotient  will  be  the 
decimal  reqiyred. 

EXAMPLES. 
1.  Reduce  |  to  a  decimal.  2.  Reduce  -}  to  a  decimal. 

OPKRATION.  OPERATION. 

5)3,0(,C  ^Ms.              The  numerator  in  these  7)l,0(,1428+^ns. 

3  0                         operations  is  considered  as  7 

an  integer,  and  always  re-  ■  ■■ 

0  G                         quires  the  decimal  point  to  30 

be  placed  imtnodiately  af-  28 

ter  it,  the  cyphers  annexed  occupy  the  places  — ! —    * 

of  decimals,  the   quotient  must  be  pointed  off  20 

accord iflg  to  the  rule  in  division.  14 

60 
56 


3.  Reduce  -! ,  \,  and  4^  to  dec:mals.     Answers,  ,2p    ,5.  ,76. 


\ 


4.  Reduce  >,-,  ^{,j,  and  jf^^  to  decimals.         Ans.  ,1923+,02o  ,00797^-1 


CASE  2. 

To  reduce  nwnhers  of  different  denominations,  as  of  Money,  Weight  and 

Measure  to  tJieir  decimal  values. 

RULE. 
"  I.  Write  the   given  numbers  pfjr[)endicularly  under  each  other  for 
dividends,  proceeding  orderly  from  the  least  to  the  greatest. 


Sect.  II.  2.  DECIMAL  FRACTIONS.  75 

"  II.  Opposite  to  each  dividend  on  the  left  haiuT;  place  such  a  number 
•  lor  a  divisor  as  will  bring  it  to  the  next  superior  denomination  and  draw 
'  a  line-perpendicularly  between  them.         . 

"  III.  Begin  with  the  highest  and  write  fiie  quotient  of  each  division,  as 
'•  decimal  parts  on  the  right  band  of  the  dividend  next  below  it,  and  so  on, 
'•  ;ill  they  are  all  used,  and  the  last  quotient  will  be  the  decimal  sought." 

EXAMPLES. 
1.  Reduce  10s.  G|c?.  to  the  fraction  of  a  pound. 

The  given  numbers  arranged  l^r  the  op- 
eration, all  stand  as  intogers.     I  then  sup- 
pose   2    cyphers    annexed    to    the  3(.3,00; 
which  divided  by  4,  the  quotient  is  75,  which 
,628125  Aim.     I  write  against  six  in  the  next  line,  and  the 
sum  thus  produced  (0,76)  I  divide  by  12,  placing  the  quotient,  (5625)  at 
the  right  hand  of  the  10;  lastly,  1  divide  by  20  and  the  quotit^nt  (,528125) 
is  the  decimal  required. 


2.  Reduce  13s.  6Jrf.  to  the  deci-         3.  Reduce   I2pri-ts.    14g rs.  to   the 
mal  of  a  pound.     Aiis.  ,6729+  decimal  of  an  ounce.     Ans.  ,6291, 


OPERATIOr.. 

4 

3, 

12 

6,75 

20 

10,6625 

CASE  3. 

To  find  ihe  value  of  any  given  decimal  in  the  terms  of  an  integer. 

RULE. 
Multiply  the  decimal  by  that  number  which  it  takes  of  the  next  less  de- 
nomination to  piake  one  of  that  denomination  in  which  the  decimal  is  given, 
and  cut  oil  so  many  ligures  for  a  remaindor  to  the  right  liai.d  of  the  quo- 
tient, as  there  are  plates  in  the  given  decimal.  Proceed  in  the  same  man- 
ner wilh  (he  remainder,  and  continue  to  do  so  through  all  the  parts  of  the 
integer,  aud  the  several  <3«'nominalions  i.tandirg  on  the  lef:  hand  make  the 
answiT 


76  DECIMAL  FRACTIONS.  Sect.  II.  2. 

EXMIPLE'i. 

1.  What  is  the  value  of  .528125  of 
a  pound  ? 

OPERATION'.  This  question  is  the  fust  example 

,528125  in  the  preceding  case  inverted,  hy 

2  0'  which  it  will  be  seen  that  questions 

in  these  two  cases  may  reciprocally 

prove  each  other. 

The  given  ilecimal  being  the  deci- 
mal of  a  pound,  and  shillings  being 
the  next  less  inferior  denomination, 
because  20  shillings  make  one  pound, 
I  multiply  the  decimal  I>y  20,  and  cnt- 
FarthiiiL(s    3,0  0  0  0  0  0  ^    ting  off  from  tl:e  right'  hand  of  the 

.i7is.  lOs.  6^d.  product  a  number  of  figures,   for  a 

remainder  equal  to  the  number  qi 
figures  in  tlie  given  decimal,  leaves  10  on  the  left  liand  which  are  shillings. 
i  then  multiply  the  remaindet",  which  is  the  decimal  of  a  shillirig  by  12,  and 
cutting  ofif  as  before,  gives  6  on  the  left  hand  for  pence  ;  lastly.,  I  multiply 
this  last  remainder,  or  decimal  of  a  pennj'  by  4,  and  find  it  to  be  3  fnrthings, 
without  any  remainder.  It  then  appears  that  552t>125  of  a  pound  is  in  va- 
lue 10s.  6id. 


2.  What  is  the  value  of  ,73968  of        3.  Vvhat  is  the  value  of  ,768  of  a 
a  pound  ?  Jlns:  14.?..  9^d..  pound  Troj  ? 


Sliillings  1 

0,5 

G 

o 

6 

0 
1 

0 

a 

Fence 

G,7 

5 

0 

0 

0 

0 
4 

*  II  is  the  last  remainder,  G80  reduced  to  its  lowest  ternw.  A  fraction 
is  said  to  be  reduced  to  its  lowest  terms,  when  there  is  no  number  which 
will  divide  both  the  numerator  and  denominator  without  a  remainder. — 
Thus,  set  to  the  fraction  its  proper  denominator  tVo'V'  then  divide  the  nu- 
merator and  the  denominator  by  any  number  v/hich  will  divide  them  both, 
without  a  remainder,  continue  to  do  .so  as  long  as  any  number  can  be  found 
that  v/ill  divide  them  in  that  manner. 

4. 

Q'S  _8J>JL S  -,    —s  1 7 

"y  I  0  o  0  i-ji  2T' 


Sect.  II.  2.  SUPPLEMENT  TO  FRACTIONS.  7T 

SUPPLEMENT  TO  FRACTIONS. 


QUESTIONS. 

1.  Wh?ct  are  fractions? 

2.  What  are  iiilogerd  or  Avhole  nuuibcrs  ? 

3.  What  arc  mixed  riuiubcrj  .' 

4.  Of  how  many  kiml^  arc  fnictions  ?  : 
3.  ifow  are  Vulgar  Fraclioiis  written?                                                                     m 
6. ,  What  is  signified  by  the  denominator  of  a  fractioa?  j 
7,  What  is  eignifiud  by  the  numerator  ? 
G.  How  are  Decimal  Fractions  written  ? 
0.  How  do  Decimals  tlifl'er  from  Vulgar  Fractions  ? 

JO.  How  can  it  be  ascertained  what  the  denoaiinator  to  a  Decimal  Frae- 
tion  is  if  it  be  not  expressed  ? 

11.  How  do  cyphers  placed  at  the  left  hand  of  a  Decimal  Fraction  affect 
its  value  ? 

12.  How  are  Decimals  distinguished  from  whole  numbers  ? 

13.  In  the  addition  of  Decimals  what  is  the  rule  for  pointing  off? 
M.  What  is  the  rule  of  pointing  off  Decimals  in  Subtraction  ?  In  Multi- 
plication ?  aiid  in  Di\  ision  ? 

15.  In  what  manner  is  the  reduction  of  a  Vulgar  Fraction  to  a  decimal 

performed  ? 
IG.   How  are  numbers  of  di (To rent   denominations,  as  pounds,  shillings, 

pence,  &.c.  reduced  to  their  decimal  values  ? 
17.  If  it  be  required  to  find  tiie  value  of  any  given  decimal  in  the  terms 

of  an  integer,  wliat  is  the  method  of  procedure  ? 

EXERCISES. 

1.  What  is  the  sum  of  79]-  6i  and         In  Case  1.  Ex.  3d,  under  Ftcd-ic- 
of -i?  when  added  together. 

GPEiiATio.v.  lion  of  decimal  fractions,  the  Scholar 

79,5 

G,25  may  notice  tliat  i,  J  and  ■}  reduced 

,75 

to    decimals    are,    ,25,   ,5  and   ,7^. 

When  numbers,  therelbre,  fur  ope- 

2.  From  17  take  f 
orcRATioN.  rations  in  either  of  the  fundamental 

17,  _ 
s^S  Ptulcs,  are   incumbered  with  these 


86,60  Ans. 


16,25  Rsmainder.  fractions  I,  ■].  {,  substitute  fjr  them 


sXrPLEMENT  TO  FRACTIONS. 


Sect.   II.  2- 


3.  Multiply  GQ\  by  5J. 

OPERATION. 
6    8,  2    5 

5,  5 


3  4   12  6 

3  4   12  5 


3  7  5,  3  7  5  Product: 
4.  Divide  26a  by  ^. 

OPERATION. 

2,5)26,25(10,5  Qtuotieitl. 
25 


125 
125 


"ill 

their   equivalent  decimal  fractions,  ^! 
(hat  is,  for  i  ,25  for  •}  ,6  ibr  3  ,75 
then  proceed  according  to  the  rule^ 
already  given  for  these   respectiva^   | 
operations  in  decimal  fractions. 


Many  persons  arc  perplexed  by  occurrences  of  a  similar  nature  to  the 
examples  above.  Hence  is  seen  in  some  measure  the  usefulness  of  frac- 
tions, parlicuiarh-^  decimal  fractious.  The  only  thing  necessary  to  render 
an}'^  person  adroit  in  these  operations  is  to  have  riveted  in  his  mind  the 
rules  for  pointing  as  taught  and  explained  in  their  proper  places.  They  are 
not  burthensome  ;  every  scholar  should  have  them  perfectly  committed. 

6.  If  a  pile  of  wood  be  18  feet  A  cord  of  wood  is  128  solid  feet ; 
long,   1 H  wide,  and  7-^^  high,  how     the  proportions  commonly  assigned 


many  cords  does  it  contain  ? 
^7nr.  12corrf.^.  08 />"?/•  43: 


are,  8  feet  in  length,  4  in  breadth, 
and  4  in  height. 

The  contents  of  a  load  or  pile  of 
wood  of  any  dimensions  may  be  found 
by  multiplying  the  length  by  the 
breadth,  and  this  product  by  the 
height  ;  or.  by  multiplying  the 
length,  breadth  and  height  into  each 
other.  The  last  product  divided  by 
128  will  shew  the  number  of  cords, 
the  remainder,  if  any,  will  be  so 
many  solid  feet.  > 


*  The  432  inches  is  the  frsction,  .25  of  n  foot,  valnrd  acrordhng  {n  C..Kf,x  3,  deduction 
Decimal  Fractions. 


Sect.  II.  2.  SUPPLEMENT  TO  FRACTIONS.  79 

6.  If  a  load  of  wood  be  9  feet  long,         7.  What  is  the  value  of  ,725  of  a 
3i  feet  wide,   and  4  feet  high,  how     day?  Arts.  11  hrs.  2\min. 

many  square  feet  does  it  contain  ? 

Ans.  126  feet,   •a.'hich  are  in,'o  feet 
short  of  a  cord. 


8.  What  is  the  value  of  ,0625  of  a        9.  Reduce  SCut.  Ogrs.  7/6.   Boz. 
shilling  ?  Ans.  3  farthings.       to  the  decimal  of  a  ton. 

Ans.  ,  1 533482 1-f 


10.  Reduce  3  farthings  to  the  deci-        11.  Reduce  j|y  to  a  decimal  frac- 
mal  of  a  shilling?  ./??!s.  ,0625.  tion.  Ans.  fil2b. 


€0  FEDERAL  MONEY.  Ssct.  II.  3. 

^  tJ.  FEDERAL  MONEY. 


Federai.  I^foNEY  is  the  coin  of  the  United  States,  established  by  Con- 
gress, A.  D.  lliiG.  Of  all  coius  this;  is  the  most  simple,  aud  the  operatious 
iu  it  the  most  easy. 

Tho  denoDiinaticns  arc  in  a  den'mal  proportion,  as  exhibited  in  the  fol- 
Icwiug 

TABLE. 

10  Mills         \  r   Cent, 

10  Cents        \     .u         r,f    Dime, 

10  Dimes       C  "^   i    Dolhv,  marked  thus,  ^ 

10  Dollars     )  (    Eagle. 

Tlie  expression  of  any  sum  in  Federal  Money  is  sinrply  the  expression  of 
a  mi.itd  number  in  decimal  li-actions.  A  dollar  is  the  Unit  Money ;  dollars 
thejol'ore  nu'.st  occupy  the  pluCe  of  units,  the  less  dcnotainalions,  as  dimes, 
cents,  and  mills,  are  de<:i;ual  parts  of  a  dollar,  and  may  be  distinguished 
from  dollars  in  the  same  way  as  any  other  decimals  by  a  comma  or  separa- 
trix.  AH  the  figures  to  the  left  hand  of  dollars,  or  beyond  units  place  are 
ea;^les.  Thus,  17  eagles,  6  dollars,  3  dimes,  4  cents,  and  6  mills  are 
written — 


Of  these,  four  are  real  coins,  and  one  is  imaginary. 

The  real  coins  are  the  Eagle,  a  gold  coin  ;  the  Dol- 
lar and  the  Dime,  silver  coins  ;  and  the  Cent,  a  copper 
coin.  The  Jlill  is  only  imaginary,  there  being  no  piece 
of  money  of  that  denomination. 

There  are  half  eagles,  half  dollars,  double  dimeg, 
half  dimes,  and  half  cents,  real  coius. 


These  denorainalions,  or  different  pieces  of  money,  being  in  a  tetifold 
proportion,  consequently  any  sum  in  Federal  Money  does  of  itself  exhibit 
the  particular  number  of  each  diiTcrent  piece  of  money  contained  in  it. 
Thus,  175,346  {seventeen  eagles.  Jive  dollars,  ikree  dimes,  four  cents,  six  mills) 
contain  1753:0  mills,  17534  ^%  cents,  1753  -^Vo  dimes,  175  V^.W  dolls. 
17  -,VW'd  tagles.  Therefore,  eagles  and  dollars  reckoned  together,  ex- 
press the  number  of  dollars  contained  ia  th«i  sum  ;  the  same  of  dimes  and 
cents ;  and  thic  indeed  is  the  usual  way  of  account,  to  reckon  the  wkole 
sum  in  dollars,    cents,  and  mills,  thus  : 

$175-34  6 

The  Addition,  Subtraction,  Mulliplication  and  Division  of  Federal  Money 
is  perlbrnu'd  in  all  respecis  as  in  Decimal  Fractioiis,  to  which  the  Scholar 
is  referred  for  the  use  ufi-ul*^^  in  these  oueratious. 


■a 

^  -a  -2 

C! 

■5    m 

3  ^V 

^  -?;  -o  '^ 

-3  r** 

'c,^  5  S 

o 

^^..'-^g 

fcT  t-  .'•    ^ 

irj 

c    0    3    f^ 

O 

0 

-:!:  2  ^  '^^ 

w 

-o.E  5  = 

".^v^. 

k— 1  wJ  '^^  ct; 

1    7 

5,  0  4  tj 

5ect.  II.  3.         ADDITION  OF  FEDERAL  MONEY.  81 

ADDITION  OF  FEDERAL  MOJVEY. 
1.  Add  16  Eagles  ;  3  Eagles,  7  Dollars  5  Cents  ;  26  Dollars,  6  Dimes, 
4  Cents,  3  Mills  ;  75  Cents,  8  Mills,  40  Dollars,  9  Cents  together. 

OPERATION.  2.  If  I  am  indebted  59  dollars,  112 

^  .  .                         dollars,  98  cents,  113  dolls.  15  ct3.  15 

to     ^      J  «  ?i                        dollars,  21  dollars,  60  cents,  200  dol- 

l^      q      Q  Cj  ^                        lars,  73  dollars,  35  dollars,  17  cents, 

^--^^  75  dollars,   20  dollars,  40  dollars,  33 

16      0,  cents  and  16  dollars.    What  is  the  sum 

3  7,      0  5  which  I  owe?            Am.  %n\   13. 
2      6,      6  4  3 

,      7      5       8 

4  0,      0      9 


4, 


Or  the  sums  may  be  all  reckoned 
in  dollars,  cents  and  mills,  thus, 


15       § 

i^ 

^160 

37  05 

26  64 

3 

75 

8 

40  09 

g264  54     1     • 

Accountants  generally  omit  the  comma,  and  distinguish  cents  from  dollars 
by  setting  them  apart  from  the  dollars. 


SUBTMCTIO^:  OF  FEDERAL  MO^rEY. 
1.  From  ^863,  17  take  $69,  82.  2.  From  ^681  take  ^57,63, 

OPERATioif.  Remainder,  ^623,37. 

8  6  3,  1  7 
6  9,  8  2 


Remainder,  7  9  3,  3  6 


8e 


MULTIPLICATION  OF  FEDERAL  MONEY     Sect.  II.  3. 


MULTIPLICATION  OF  FEDERAL  MONEY. 
1.  If  flour  be  glO,25  j>€r  barrel,  what  will  27  barrels  cost  ? 

OPERATIO\. 

1  0,  2  5 


2  7 


2  0 


$2  7  6,  7  bAns. 

2.  Multiply  ^76,35  by  $37,46. 
Prodvct,  $20GO,O71O. 


Point  off  the  decimals  in  the  pro- 
duct according  to  the  rule  in  Multi- 
plication of  decimals  ;  if  at  any  time 
there  shall  be  more  than  three  de- 
cimal figures,  all  beyonc^mills  or  the 
third  place,  will  be  decimal  parts  of 
a  mill. 

3.  Multiply  $24,675  by  g  13,63, 
Product,  $336,320yV-o- 


DIVISION  OF  FEDERAL  MONEY. 
1.  If  2728  bushels  of  wheat  cost  $2961,  how  much  is  it  per  bushel? 

OPERATION. 


Bushels.  Dolls.  D. 
2728)2961(1, 
2728 


m. 

5  Ans.. 


23300 
21824 

14760 
13640 

1120 

2.  Divide  $3766  equally  among 
13  men  ;  what  will  each  man  re- 
ceive ?  Ans.  $288,923. 


When  the  dividend  consists  ot 
dollars  ottly,  if  there  be  a  remain- 
der after  division,  cyphers  must 
be  annexed  as  in  division  of  deci- 
mals. 


Divide  $16,75  by  27. 
Quotmit,  62  cents. 


Sect.  II.  3.      SUPPLEMENT  TO  FEDERAL  MONEY.  83 

SUPPLEMENT  TO  FEDERAL  MONEY. 


1. 


QUESTIONS. 
What  is  Federal  Money  ?  When  was  its  establishment,  and  by  what 
authority  ? 

2.  What  are  the  denominations  in  Federal  Money  ? 

3.  Which  is  the  Unit  Money  ? 

4.  How  are  dollars  distinguished  from  dimes,  cents,  and  mills  ? 

5.  What  places  do  the  different  denominations  occupy,  from  the  decimal 

point  ? 

6.  How  is  the  addition  of  Federal  Money  perforn»ed  ?   Subtraction  ? 

Multiplication  ?    Division  ? 

EXERCISES. 
1.  A  man  dies,  leaving  an  estate 
of  ^71600,  there  are  demands  against 
the  estate  of  ^39876,74  ;  the  residue 
is  to  be  divided  between  7  sons  ; 
what  will  each  one  receive  ? 


Ans.  ^4331  Zdcts, 


2.  A  man  sells   1225  bushels  of 
wheat  at  $1,33  per  bushel,  and  re- 
ceives   ^93,76    for    transportation ; 
what  does  he  receive  in  the  whole  ? 
w5?M.  $1723,01. 


3.  What  will  3  hogsheads  of  sugar 
cost,  each  weighing  3Ca'<.  ^qrs.  lib. 
at  16crs.  1  mills  per  lb.  ? 

Ans.  gI99,899. 


4.  Divide  seven  tJiousand  six  dol- 
lars, one  cent  and  three  mills,  by 
five  hundred  seventy  six  dollars, 
thirty  four  cents  and  tv.'o  mills. 

Ans.  $13,155.. 


84  EXCHANGE.  Sect.  II.  4. 

^  4.  EXCHANGE. 


Exchange  is  the  giving  of  the  bills,  money,  weight,  or  measure  of  one 
place  or  country,  for  the  like  value  in  the  bills,  money,  weight  or  measure 
of  another  place  or  country. 

Norn  1.  The  Currencies  in  the  >c\v  EnglanJ  States,  and  in  Virginia,  are  the  same 
and  \vili  be  all  comprehended  under  the  term  A".  E.  airreJtcr/ ;  those  of  ^ew- York,  North 
Carolina  and  Ohio,  are  the  same,  and  will  be  comprehended  under  tlie  term  JV".  York 
Ciirrenni ;  thoj^e  of  N.  Jersey,  lennsj  Ivania,  Delaware  and  Maryland,  are  the  same,  and 
will  all  bo  comprehended  under  the  tcnn  Fenn.  Currency. 

Note  1.  It  will  be  sufficient  perhaps  in  most  cases,  that  the  pupil  be  re<iuired  to  work 
Uibrule  in  the  cunency  of  that  Slate  only  to  which  he  belongs. 

CASE   1. 
To  dian^e  Xc-ji- England,  ^c.  and  A eu-  York,  i'C.  Currencies  to  Federal  Money. 

RULE. 
Set  tlou-n  the  pounds,  and  to  the  right  hand  write  half  the  greatest  even 
niinibor  of  the  given  shillings  :  then  consider  how  many  farthings  there  are 
contained  in  the  given  pence  and  farthings,  and  if  the  sum  exceed  12,  in- 
crease it  by  1,  or  if  it  exceed  36,  increase  it  by  2,  which  sum  set  down  to 
the  right  ha:id  of  half  the  greatest  even  number  of  shillings  before  written, 
remembering  to  increase  the  second  place,  or  the  place  next  to  shillings 
l»y  5,  if  the  shillings  be  an  odd  number ;  to  the  whole  sum  thus  produced, 
annex  a  cypher,  and  divide  the  sum  by  3,  if  it  be  JV.  England  currency, 
and  by  4  if  it  be  A'ew-York ;  cut  off  the  three  right  hand  figures  in  the 
quotient,  which  will  be  cents  and  mills  ;  the  rest  will  be  dollars, 

EXMIPLES. 
1.  Change  £47  7s.   lO^d.  to  dollars,  cents  and  mills. 

OPERATION. 

^  J:  6^  In  this  example  to  the  right  hand  of  pounds 

■5  rt  2  ("^~)  ^  write  3,  half  the  greatest  even  number 

^-a  2  of  the  given  shilhngs(7)  ;  the  farthings  in  lO^d. 

^  (43)   increased   by  2  (45)   because    exceeding 


O    0)  • -2 


rfi 


36    and    the    second  place  increased  by  6  be- 
cause the   shillings  were  an  odd  number,  make 
S  ^  ^  T3    05,  which  sum  written  to  the  right  hand  of  the 
c  •-  'a  ^    3,  a  cypher  annexed,  and  the  sum  divided  by  3 
•ij     S  ^«  a    gives  the  answer  157  (/o//ar5,  98  cen^s,  and  3  ;/u7/s 
1     tj  IS  u  ri    for  JV.  England  currency;  the  same  sum  (473950) 
divided  by  4,  gives  118  dollars,  48  cents,  7  mills 
for  JV.  York  currency. 


Ci-   -r; 


J  ti«  .a  o) 


Divide  by 3) 4  7  3  9  5  0 


Dolls.   1   5  7,  9  8  3 

If  there  be  no  shillings,  or  only  1  shilling  in  the  E:iien  sum,  so  there  be  no  even  numberf 
write  a  cvpher  in  plai.e  of  half  the  even  rmmber  of  shillings,  then  proceed  with  the  pence 
and  fartliingsas  in  other  cases. 

If  pounds  only  are  friven  to  be  changed,  annex  a  cypher  and  divide  as  before,  the  quotient 
will  Ije  dn'Iars.  If  tii*^re  be  a  remainder,  annex  3  wore  cyj)hers  and  divide,  the  quotient 
will  if  ciiiiu and  mills. 

If' yo'iiuh  and  an  even  number  of  shillings  only  be  giren,  to  the  pounds  annex  half  th* 
f  vcn  nunsbcr  of  siiilliiigs,  divide  as  before,  and  the  quolierit  will  bo  dollars. 

A  little  priictice  wiilaiakc  llicse  operatioris  exifcmely  easy. 


Sect.  II.  4.  EXCHANGE.  86 

2.  In  £763  JV.  E.  and  JV.  F.  cur-  3.    In  ~£17  Is.  B^d.   how  many 

rencies,  how  many  dollars,  cents,  and  dollars,  cents  and  mills  ? 

mills  ?  Ans.  $66  92  3.  jY.  E.  cur, 

Ans.  ^2543  33cts.  3m.  A'*.  E.  cur.  42  69  2.  JV.  Y.  — 
1907  60  JV.  Y.  — 


4.  In   £109    35.   8c?.    how  many  5.  In  £86  Gs.  Sjc?.    how   many 

dollars  and  cents  ?  dollars,  cents  and  mills  ? 

Ms.  $363,94  N.  E.  cur.  Ans.  $287,740  A".  E.  cur. 

272,95  A".  Y.  —  215,805  A^  Y.  — 


6.     Exchange   £l    \s.   lO^d.    to  7.  Exchange  £10  4ii.  to  rezo' 

Federal  Money.  ral  Money. 

Ans.  $3,646  N.  E.  cur.  Ans.  $33,396  A^.  E.  cur. 

2,735  A".  Y.  —  25,047  A".  Y.  — 


8.    Exchange    £103   to   Federal         9.     Exchange   ^d.    to    Federal 

Money.  Money. 

Ans.  $343,333  A'.  E.  cur.  Ans.  Sets.  Gm.  X.  E.  cur. 

257,50   A".  Y.  —  2—   7—  A".  Y.  — 


CASE  2. 

'To  Exchange  Federal  Money  to  New-England  and  New-York  Curtencies. 

RULE. 

''  If  there  be  no  mills  in  the  given  sum,  reduce  it  to  mills  by  annexing  cy- 
phers ;  multiply  the  given  sum  by  3,  if  it  be  required  to  change  it  to  N.  E. 
curreVcy  ;  but  if  to  the  currency  of  N.  York,  by  4  ;  cut  off  the  four  right 
hand  figures,  which  will  be  decimals  of  a  pound,  the  left  hand  figures  will 
be  the  pounds.  To  find  the  value  of  the  decimals,  double  the  first  figure 
for  shillings,  and  if  the  figure  in  the  second  place  be  5,  add  another  shilling, 
then  call  the  figures  in  the  second  and  third  places,  after  deducting  the  5 
in  the  second  place,  so  many  farthings,  abating  1  when  they  are  above  12, 
and  2,  when  tliey  are  above  36.  ^ 


86 


EXCHANGE. 


Sect.  II.  4. 


•  EXAMPLES. 

1.  Cbang'e  255  dollars,  40  cents,  6  mills,  to  pounds,  shillings,  pence  and 
f/irlhiDgs. 

OPERATION.  OPER^TIOy. 

255406  25540G  Having  multiplied  and  cut 

3  4  oft"  the   lour  riglit  hand  fig- 

ures as  the  rule  directs,  to 
llnd  the  value  of  the  figures 
cut  off,  I  double  the  first 
figure  (G)  A'.  E.  cur.  which 
gives  r2  for  shillings  ;  the 


G|6218  102|162l 


.^na.  £76   12s.   bd. 
.V.  E.  cur. 


£102  3s.  3(7. 
.'V.  Y.  cur. 
figures  in  the  second  and  third  places  (21)  abating  1  for  being  over  twelve 
(20)  are  to  be  considered  so  many  farthings,  which  reduced  to  pence  are  5. 

THr  3s.  3d.  N.  Y.  cur.  are  obtained  after  the  same  manner.  The  dou- 
ble of  the  first  figure  cutoff  (1)  is  2,  and  because  the  figure  in  the  second 
place  (6)  is  more  than  6,  1  add  another  shilling,  making  3s.  then  the  figures 
ill  the  second  and  third  places  (62)  after  deducting  the  5  for  1  shiUing  from 
the  G,  are  12,  which  reduced  to  pence  are  3. 

The  o  and  the  4  in  the  fourth  places,  being  something  less  than  one  far- 
thing, are  lo.st,  not  being  reckoned. 

If  there  he  neither  cents  nor  mills,  that  is,  if  the  given  sum  be  dollars,  mul- 
tiply by  3  and  cut  off  one  figure  only. 

2.  In  §392, 75  how  many  pounds,  3.    In  ^39,635  how  many  pounds, 

shillings,  pence  and  farthings  ?  shillings,  pence,  &.c.  ? 

Ans.  £117  IGs.  &d.  N.^E.  cur.  Ms.  £11    17s.  9irf.  JV.  E.  cur. 

157    2     0    A*.  F.    —  15  17      r     N.  Y.  cur. 


4.  Exchange  134  dollars  65  cents 
to  pounds,  shillings,  pence  and  far- 
things. 

Ans.  £40     7s.   lO^d.  N.  E.  cur. 
63  17        2^     JV.  Y.  cur. 


5.  Exchange  684  dollars  to  pounds 
and  shillings. 

Ans.  £205  4s.  JV.  E.  cur. 
273  12  JV.Y.  cur. 


6.  Exchange  71  cents  to  shilhngs, 
pence,  &c. 

Anx.  4.».  3^.  .V.  E.  cur. 
5     8 1    .Y.  Y.  cur. 


7.  Exchange   13cfs.  1m.  to  pence       • 
and  farthings. 

.Ins.  ':^l(l.  N.  E.  cur. 
\3d.  K.  Y.  cur.  ' 


Sect.  II.  4.  EXCHANGE.  87 

CASE  3. 

To  change  New-Jersey,  Pennsylvania,  Delaware  and  Maryland  Currency  to 

Federal  Money. 
RULE. 
Reduce  the  given  sum  to  pence,  annex  a  cypher,  divide  these  pence  by 
9,  and  add  the  quotient  to  the  pence  ;  from  the  sum  point  off  three  figures, 
which  will  be  cents  and  mills ;  those  to  the  lellt  hand  will  be  dollars. 

If  there  are  farthings  in  tlie  given  sum,  in  place  of  the  cypher  annex  2  for 
1  fartuing  ;  5  for  2  farthings  ;  7  for  3  farthings,  and  proceed  as  before. 

Jf  the  ^ivcn  sum  be  pounds  otily,  multiply  hy  8,  annex  3  cyphers  to  the 
product,  and  divide  by  3  ;  the  quotient  will  be  the  answer,  pointing  off  the 
three  right  hand  figures  for  cents  and  mills. 

EXAMPLES. 
1    Change  £17  Is.  G^d.  to  Federal  Money. 

20  I  first  reduce  the  given  sum  to  pence,  to 

— —  which   (4098)  I   annex  the  figure   5  for 

341  the  irf.  and  divide  by  9  ;  the  quotient  add- 

12  ed  to  the  pence  and  th€  three  right  hand 

figures  pointed  off  give  the   answer,  45 


9)40985  dollars,  63  cents  and  8  mills 

4653* 


Jlns.  45,538 

*  This  quotient  figure  (3)  might  with  propriety  have  been  put  down  4,  the  9's  in  35 
comiug  so  near  producing  it,  and  it  would  have  been  nearer  the  true  value  ;  the  mills  in 
the  answer  would  then  have  been  9  in  place  of  8.  , 

2.  In  £109  35.  8d.  how  many  dol-      3.  Change  £736  to  Federal  Money, 
lars,  cents  and  mills  ?  Ans-.  g  1962,666. 

Ans.  ^291,155. 


4.  In   £8G    6s.  5i(Z.   how   many       i.  Ch«ige  G^d.  to  Federal  Money, 
dollars,  cents  and  mills  ?  Ans.  lets.  4  mills. 

Ans.  $230,191. 


88  EXCHANGE.  Sect.  II.  4. 

CASE  4. 

To  thange  Federal  Money  to  Nen:-Jersey,  Pennsylvania,  Dda'ware  and  Mary- 
land Currency. 

RULE. 
If  there  be  no  mills  in  the  given  sum,  reduce  it  to  mills  by  annexing  cy- 
phers, subtract  one  tenth  of  itself,  the  remainder,  except  the  right  hand 
lip^ure,  will  be  pence,  which  must  be  reduced  to  pounds  ;  to  find  the  value 
of  the  right  hand  figure,  if  it  be  2,  reckon  1  farthing  ;  if  5,  reckon  it  2  far- 
things ;  if  7,  reckon  it  3  farthings. 

I^'oTE. — Subtracting  the  tenth  of  the  given  sum  from  itself  may  be  done  in  this  man- 
ner : — Suppose  the  sum  6452.    Write  tiie  given  sum  under  itself,  removing 
6  4  5  2    the  figures  one  place  towards  the  right  hand  and  dropping  the  right  hana 
6  4  5    figure  ;  subtract  and  the  remainder  will  be  the  sum  required. 

6  8  0  7 

EXAMPLES. 
1.  Change  ^45,538  to  pounds,  shillings,  pence  and  farthings. 

OPERATION.  This  is  the  first  example  in  the  former  Case 

45,538  inverted.     Having  subtracted  one  tenth  of  the 

4,653  given  sum  from  itself  in  the  manner  directed  in 

^ — >  the  note  above,  the  right  hand  figure  in  the  r«- 

12)4098|5  mainder  (5)  being  to  be  reckoned  2  farthings,  I 

set  it  down  in  the  answer  ^d. — the  other  figures 

2|0)34[1  of  the   remainder  (4098)  being  pence,  I  divide 

Jins.  £,n  Is.  6^d.  by  12,  in  doing  which  there  is  a  remainder  of  6, 
which  are  pence ;  these  I  also  set  down  in  the 
answer.  The  shillings  (341)  divided  by  20,  cutting  off  one  figure  from 
the  divisor  and  one  from  the  dividend  as  is  usually  practised  in  reducing 
shillings  to  pounds,  give  £17,  and  the  1  cut  off  from  the  dividend  is  1 
shilling,  which  completes  the  answer. 


2.  Change  gl35  to  pounds,  &c.  3.  Change  ^287,74  to  pounds 

Ms.  £.50  12s.  6d.  Ans.  fA^l  18s.  Q\d. 


Sect.  II.  4.  EXCHANGE.  89 

To  change  the  Nexs: -England  to  the  New-Yorh currency ;  add  one  third. 

To  change  the  JVew-York  to  the  JVe-so-England  currency;  subtract  one 
fourth. 

To  cfutnge  the  JVew-England  to  the  Pennsylvania  currency  ;  add  one  fourtli. 

To  change  the  Pennsylvania  to  the  Neva-England  currency ;  subtract  one 
fifth. 

To  change  the  New-York  to  the  Pennsylvania  currency  ;  subtract  one  six- 
teenth. 

To  change  the  Pennsylvania  to  the  New-York  currency?  add  one  fifteenth. 


SUPPLEMENT  TO  EXCHANGE 

QUESTIONS. 

1 .  What  is  Exchange  ? 

2.  How  do  you  change  N.  England  2.  How  do  you  change  Pcnnsyl- 
and  Virginia  currencies  to  Fed-  vania,  &,c.  currency  to  Fcde- 
eral  Money  ? — New-York  cur-  ral  Money  ? 

rency  ? — and  wherein  consists 
the  difference  ? 

3.  If  pounds  only  are  given  to  be  3.  If  there  are  farthings  in  the 
changed,  how  do  you  proceed  ?  given  sum,  how  do  you  proceed  ? 

4.  When  there  are   no  shillings,  4.  If  the  given  sum  be    pounds 
or  only  one  in  the  given  sum,  only,  how  do  you  proceed  ? 
how  do  you  proceed  ? 

5.  How  do  you  change  Federal  5.  How  do  you  change  Federal 
Money  to  N.  England  currency  ?  Money  to  Pennsylvania,  &c. 
N.  York  currency  ? — Wherein  currency  ? 

consists  the  difference  ? 
G.  How  do  you  change  New-England  to  New- York  currency  ? — New 
York  to  New-England  ? — New-England   to   Pennsylvania  ? — Pennsylvania 
to  New-England  ? — New- York  to  Pennsylvania  ? — Pennsylvania  to  New- 
York  currency  ? 

M 


90  EXCHANGE.  Sect.  II.  4. 

EXERCISES. 

1.  In  £3G   Is.  61,1  N.  Enp:.  cur.  or  £48  2.v.  Oic/.  N.  York  cur.  or  £4* 
1.1.    llr/.  Penn.  rnr.  how  many  flollars,  cents  and  mills  ? 

4n5.  ^120,257  X.  E.  cur.— $l20,2oo  N.  Y.  t»r.— gl20,  256  Penn.  cur. 

NoTK. — In  making  llip  oxcliange  from  one  ciinency  into  another  tliere  will  fi-equently 
I'p  the  loss  of  snme  fiartioni  of  a  fnrfhing;  far  this  reason  when  the  exchange  is  again 
made  into  Federal  iMoney,  there  will  be  the  diflerenco  of  some  mills  in  the  answers 
obtained. 

2.  Cliange  £100  12s.  N.  E.  cur.  to  N.  Y.  cur.  Penn.  cur.  and  to  Federal 
Money. 

Ans.  £240   IGs.  N.  Y.  cur.— £225  15s.  Penn.  cur.— $602  F.  Money. 

3.  Chang:e  ^150,25  to  N.  England,  N.  York,  or  Penn.  cur.  accordingly 
as  the  pupil  may  have  been  instructed  in  one  or  the  other,  or  all  of 
these  rules. 

Jns.  £45  Is.  6d.  N.  E.  cur.— £60  2s.  N.  Y.  cur.— £56  6s.  IQic^.Penn.  eur. 

4.  Let  the  pupil  be  required  to  change  the  sums  in  New-York  and  in 
Pennsylvania  currency  in  the  above  answer,  to  New-England  currency ; 
tl)e  same  in  New-England  and  in  New- York  to  Pennsylvania  currency  ; 
and  the  same  in  New-England  and  Pennsylvania  to  New- York  currency, 
the  answers  of  which  will  reciprocally  prove  each  other. 

5.  Change  «>345,625  to  N.  Eng.  or  N.  York,  or  Penn.  currency. 

Ans.  £103  iSs.  8^d.  N.  E.  CMr.--£l38  5s.  N.  Y.  cur.— £129  12s.  2|d. 
Penn.  currency, 

6.  Change  75  cents  into  N.  E.  or  N.  Y.  or  Penn.  cvr. 

Ans.  4s.  6d.  N.  E.  cur. — 6s.  N.  Y.  cur. — 5s.  l^d.  Penn.  cur. 

7.  Change  £45  Is.  6d.  N.  E.  cur.  or  £60  2s.  N.  Y.  cur.  or  £56  6s.  \0\d. 
Penn.  cur.  to  Federal  Money.  An.s.  ^150,25. 

8.  Change  4s.  6d.  N.  E.  cur.  or  6s.  N.  Y.  cur.  or  5s.  "i^d.  Penn.  cur- 
rency to  Federal  Money.  Aas.  75  cents. 

9.  Change  £46  10s.  6icZ.  considered  in  either  currency  to  Federal  Money. 
,.      Ans.  ^155,09  N.  E.  cwr.—^l  16,317  N.  Y.  cwr.— ^124,072  Penn.  cur. 

10.  Change  f\61  to  N.  E.  or  N.  Y.  or  Penn.  currency. 

Ans.  £50  2s.  N.  E.  cur. — £60  16s.  N.  Y.  cvr. — £62  12s.  6d.  Penn.  cur. 

11.  Let  the  pupil  be  required  to  change  the   sums  in   New- York  and 
Pennsylvania  currency,  in  the   above  answer,  to   New-England  currency,  j 
fcc.  as  in  the  4tii  exercise  above. 

12.  Change  6^r7.  to  Federal  Money. 

Ans.  9  cents  N.  E.  cur. — 6  cents  7  mills  N.  Y.  cur. — 7  cents  2  mills  Penn. 
currency. 

13.  Cliangp  £263  to  Federal  Money. 

Ans.  $1376,666  N.  E.  fwr.— ^657,50  N.  Y.  ciir.— ^701,333  Penn.  cur. 


Shct.  II.  4. 


EXCHANGE. 


&t 


TABLE 

FOR  REDUCING  NEW-ENGLAND  CURRENCY  TO  FEDERAL  MONEY. 


' 

shill. 

shiil. 

shill. 

shill. 

shiU. 

0 

1 

o 

3 

4 

5 

Pence. 

Os. 

M. 

Cts.  M 

Cts.  M. 

Cts.  M. 

Cts.  M. 

Cts.  M. 

0 

IG     7 

33     3 

50 

66     7 

83     3 

1 

1 

4 

18     1 

34     7 

51      4 

68     1 

84     7 

2 

2 

8 

19     5 

36     1 

52     8 

69     6 

86      1 

3 

4 

2 

20     9 

37     5 

54      2 

70     9 

87     5 

4 

5 

6 

22     3 

38     9 

55     6 

72     3 

88     9 

5 

7 

23     7 

40     3 

57 

73     7 

90     3 

6 

8 

3 

25 

'll      6 

58     3 

75 

91     6 

7 

9 

7 

26     4 

43 

59     7 

76     4 

93 

8 

11 

1 

27     8 

44     4 

61      1 

77     8 

94     4 

9 

12 

5 

29     2 

45     8 

62     5 

79     2 

95     8 

10 

13 

9 

30     6 

47     2 

63     9 

80     6 

97     2 

11 

15 

3 

32 

48     6 

65     3 

82 

98     6 

TABLE 

FOR  REDUCING  NEW-YORK  CURRENCY  TO  FEDERAL  MONEY. 


shill. 

shill. 

sh>il. 

shill. 

shill. 

0 

1 

2 

3 

4 

6 

Pence. 

Cts. 

M. 

Os.  M. 

Cts.  .If. 

Cts.  J\I. 

Cts.  M. 

Cts.  M. 

0 

12      5 

25     0 

37     5 

50     0 

62     5 

1 

1 

0 

13     5 

26     0 

38     6 

51     0 

63     5 

o 

2 

1 

14     6 

27      1 

39     6 

62      1 

64     6 

3 

3 

1 

15     6 

28      1 

40     6 

53      1 

65     6 

4 

4 

2 

10     7 

29     2 

41      7 

54     2 

G6     7 

5 

5 

2 

17      7 

30     2 

42     7 

55     2 

67     7 

6 

6 

2 

IG     7 

31      2 

43     7 

56     2 

68     7 

7 

7 

2 

19     7 

32     2 

44     7 

57     2 

69     7 

8 

8 

3 

20     8 

33     3 

45     8 

58     3 

70     8 

9 

9 

3 

21      8 

34     3 

46     8 

59     3 

71     8 

10 

10 

5 

23     0 

35     5 

48     0 

60     5 

73     0 

11 

11 

5 

24     0 

3G     5 

49     0 

61      5 

74     0 

To  find  by  these  Tables  the  Cents  and 
pence  under  one  dollar,  look  the  shillings 
hand  column  :  then  under  the  former,  9inil 
found  the  cci^ts  ant'  mills  sousrht. 


Mills  in  any  sum  of  shillings  and 
at  top,  and  the  pence  in  the  left 
on  a  line  with  the  latter,  will  be 


92      TABLE  REDUCING  POUNDS,  fee.  TO  DOLLARS,  &c.  Sect.  H.  4. 

TABLE 

FOR  REDUCING  THE  CURRENCIES  OF  THE  SEVERAL  UNITED  STATES  TO 

FEDERAL  MONEY. 


u.  r 


•li 


1 

2 
3 
1 

2 
3 
4 
5 
G 
7 
8 
9 
10 
11 
1 
o 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 


N.  Hamp. 

1  N.  Jersey, 

Mass. 

New-York 

Pennsylva'a 

S,  Carolina, 

Rh.  Island. 

and 

Delaware, 

and 

Conn,  and 

N.  Carolina. 

and 

Georgia. 

Vir2:iiiia. 

Maryland. 

1).  cts.  7n. 

D.  cts.  in. 

D.  els.  m. 

D.  cts.  m. 

,     3 

>     3 

,     3 

,     4 

,     '7 

,     6 

,     6 

,     9 

,   10 

,     0 

,     8 

,   14 

,   M 

,   10 

11 

,   18 

,  28 

,  21 

22 

,  36 

,  42 

,  31 

33 

,  54 

,  66 

,  42 

44 

,  71 

,  69 

,  52 

56 

,  89 

,  83 

,  62 

67 

,107 

,  97 

,  73 

78 

,125 

,111 

,  82 

89 

,143 

,125 

,  94 

100 

,161 

,139 

,104 

111 

,179 

,153 

,114 

122 

,198 

,167 

.125 

133 

,214 

,333 

,250 

267 

,429 

,500 

,375 

400 

,643 

,6G6 

,500 

533 

,857 

,C?3 

,625 

667 

1,071 

1,000 

,760 

800 

1,286 

1,167 

,875 

933 

1,500 

1,333 

1,000 

067 

1.714 

1.500 

1,125 

200 

1,929 

1,667 

1.250 

333 

2,143 

1,833 

1,375 

,467 

2,357 

2,000 

1,600 

600 

2,671 

2,167 

1,625 

733 

2,785 

2,333 

1,750 

867 

3,000 

2,500 

1,875 

2 

000 

3,214 

2,667 

2,000 

2 

133 

3,428 

2,833 

2,125 

2 

267 

3,643 

3,000 

2,250 

2 

400 

3,857 

3,167 

2,375 

2 

633 

4,071 

Sect.  II.  4.  TABLE  REDUCING  POUNDS, &c.  TO  DOLLARS,  &c.      93 

TABLE 

FOR  REDUCING  THE  CLtRRENCIES,  &c.  CONTINUED. 


New-Hamp. 

New-York, 

New- Jersey, 

S.  Carolina, 

kc.  &c.       &c. 

&c. 

6cc. 

£. 

D.  c.  m. 

D.  c.  m. 

D.  c.  in. 

D.  c.  m. 

1 

3,333 

2,5 

2,666 

4,286 

2 

6,667 

6,0 

5,333 

8,571 

3 

10,000 

7,5 

8,000 

12,857 

4 

13,333 

10,0 

10,667 

17,143 

5 

16,667 

12,5 

13,333 

21,42i) 

6 

20,000 

15,0 

16,000 

25,714 

7 

23,333 

17,5 

18,667 

30,000 

8 

26,667 

20,0 

21,333 

34,286 

9 

30,000 

22,5 

21,000 

38,671 

10 

33,333 

25,0 

26,667 

42,867 

20 

66,667 

50,0 

53,333 

85,714 

30 

100,000 

75,0 

80,000 

128,671 

40 

133,333 

100,0 

106,667 

171,429 

60 

166,667 

125,0 

133,333 

214,286 

60 

200,000 

150, 

160,000 

257,143 

70 

233,333 

175, 

186,667 

300,000 

80 

266,667 

200, 

213,333 

342,857 

90 

300,000 

225, 

240,000 

o8o, 714 

100 

333,333 

250, 

266,667 

428,671 

1     200 
f     300 

666,667 

500, 

533,333 

857,143 

1000,000 

750, 

800,000 

1285,714 

400 

1333,333 

1000, 

1066,667 

1714,286 

500 

1666,667 

1250, 

1333,333 

2142,857 

600 

2000,000 

1500, 

1600,000 

2571,429 

700 

2333,333 

1750, 

1866,667 

3000,000 

800 

2666,667 

2000, 

2133,333 

3428,571 

900 

3000,000 

2250, 

2400,000 

3857,143 

1000 

3333,333 

2500, 

2666,067 

4285,714 

TABLE 

FOR  REDUCING  FEDERAL  MONEY  TO  THE  CURRENCIES  OF  THE  SEVERAL 

UNITED  STATES. 


New-Hamp. 

New-York, 

N.  .1 

ersey. 

S.  Carolina, 

&c.  &c. 

&c. 

fcc 

,.kc. 

&c. 

DOLL.  6.S, 

DOLL.  8s. 

DOLL 

7s.  Gd. 

POLL.  As.  Gd. 

D.    cts. 

£.  s.  d.  q. 

£.  S.   d.   q. 

£•  s. 

d.q. 

£.  s.  d.  q. 

,01 

3 

1  0 

1  0 

2 

,02 

1  2 

2  0 

1  3 

1  0 

,03 

2  1 

3  0 

2  3 

1  3 

,04 

3  0 

3  3 

3  2 

2  1 

,05 

3  2 

4  3 

4  2 

2  3 

,06 

4  1 

5  3 

5  2 

3  1 

,07 

5  0 

6  3 

6  1 

4  0 

,08 

5  3 

7  3 

7  1 

4  2 

,09 

6  2 

8  3 

8  0 

6  0 

,10 

7  1 

9  2 

9  0 

6  2 

94      TABLE  REDUCING  DOLLARS,  Sec.  TO  POUNDS, &c.   Sect.  H.  4. 

TABLE 

FOR  REDUCING  THE  ClKllE.NCIES,  kc.  CONTINUED. 


Niew-Hamp. 

New-York, 

New-Jersey, 

S.  Carolina,. 

&c.  kc. 

&ic. 

&c. 

&c. 

Uvll.        cts. 

£.  s.  (/.  f/. 

£.  s.  d.  q. 

£.  s.  d.  q. 

£.  s. 

d.q. 

\,20 

1  2  2 

1  7  1 

1  6  0 

11  1 

;:30 

1  9  2 

2  4  3 

2  3  0 

1 

4  3 

,40 

2  4  3 

3  2  2 

3  0  0 

1 

10  2 

,50 

3  0  0 

4  0  0 

3  9  0 

2 

4  0 

,60 

3  7  1 

4  9  2 

4  6  0 

2 

9  2 

,70 

4  2  2 

6  7  1 

5  3  0 

3 

3  1 

,00 

4  9  2 

6  4  3 

0  0  0 

3 

8  3 

,90 

6  4  3 

7  2  2 

6  9  0 

4 

2  2 

1 

6  0  0 

8  0  0 

7  6  0 

4 

8  0 

2 

12  0  0 

16  0  0 

15  0  0 

9 

4  0 

3 

13  0  0 

1  4  0  0 

1  2  6  0 

14 

0  0 

4 

1   4  0  0 

1  12  0  0 

1  10  0  0 

18 

8  0 

5 

1  10  0  0 

2  0  0  0 

1  17  6  0 

1  3 

4  0 

6 

1  16  0  0 

2  8  0  0 

2  5  0  0 

1  8 

0  0 

7 

2  2  0  0 

2  16  0  0 

2  12  6  0 

1  12 

8  0 

8 

2  8  0  0 

3  4  0  0 

3  0  0  0 

1  17 

4  0 

9 

2  14  0  0 

3  12  0  0 

3  7  6 

2  2 

0  0 

10 

3  0  0  0 

4  0  0  0 

3  15  0 

2  6 

8  0 

20 

6 

8 

7  10  0 

4  13 

4 

30 

9 

12 

116  0 

7  0 

0 

40 

12 

16 

15  0  0 

9  6 

8 

60 

15 

20 

18  15  0 

11  13 

4 

GO 

18 

24 

22  10  0 

14  0 

0 

70 

21 

28 

26  5  0 

16  6 

8 

80 

24 

32 

30  0  0 

18  13 

4 

90 

27 

36 

33  15  0 

21  0 

0 

100 

30 

40 

37  10  0 

23  6 

8 

200 

60 

80 

75  0  p 

46  13 

4 

300 

90 

120 

112  10  0 

70  0 

0 

400 

120 

IGO 

160  0  0 

93  6 

8 

500 

150 

200 

187  10  0 

116  13 

4 

600 

180 

240 

225  0  0 

140  0 

o« 

700 

210 

280 

262  10  0 

163  6 

8 

800 

240 

320 

300  0  0 

186  13 

4 

900 

270 

560 

337  10  0 

210  0 

0 

1000 

300 

400 

375  0  0 

233  6 

8 

£000 

600 

800 

750 

466  13 

4 

3C00 

900 

1200 

1125 

700  0 

0 

40(^0 

1200 

ICOO 

1500 

933  6 

8 

5<o00 

1500 

2000 

1875 

1166  13 

4 

6000 

1800 

2400 

2250 

1400  0 

0 

7000 

2100 

2800 

2626 

1633  6 

8 

m\M) 

2400 

3200 

3000 

1866  13 

4 

9000 

27tH> 

3G00 

3375 

2100  0 

0 

lOtOO 

3000 

4000 

3750 

2333  6 

8 

SCCT.  II.  5.  SIMPLE  INTEREST. 

^  5.  SIMPLE  INTEREST. 


INTEREST  is  the  allowance  given  for  the  use  of  money,  by  the  bor- 
rower to  the  lender.  It  is  computed  at  so  many  dollars  for  each  hundred 
lent  for  a  year,  (/)«r  annum)  and  a  like  proportion  for  a  g^reater  or  less 
time.  The  highest  rate  is  limited  by  our  laws  to  6  per  cent, '^  that  is  6  dol- 
lars for  a  hundred  dollars,  6  cents  for  a  hundred  cents,  ^G  lor  aX)100,&;c. 
This  is  called  legal  interest,  and  is  always  understood  when  no  other  rate  is 
mentionGd. 

There  are  three  things  to  be  noticed  in  Interest. 

1.  The  PniNciPAL  ;  or  raancy  lent. 

2.  The  Rate  ;  or  sum  per  cent,  agreed  on. 

3.  The  Amount  ;  or  principal  and  interest  added  together. 

Interest  is  of  two  sorts,  Simple  and  Compound. 

1.  Simple  Interest  is  that  which  is  allowed  for  the  principal  only. 

2.  Compound  Interest  is  that  which  arises  from  the  interest  being 
added  to  the  principal,  and  (continuing  in  the  hands  of  the  lender)  becomes 
a  part  of  the  principal  at  the  end  of  each  stated  time  of  payment. 

GENER.iL  RULE. 

1.  For  one  year,  multiply  the  principal  by  the  rate,  from  the  product  cut 
off  the  two  right  hand  tigures  of  the  dollars,  which  will  be  cents,  those  to 
the  left  hand  will  be  dollars ;  or,  which  is  the  same  thing,  remove  the 
separatrix  from  its  natural  place  two  tigures  towards  the  loft  hand,  then  a".\ 
those  figures  to  the  left  hand  will  be  dollars,  and  those"  to  the  right  hand 
will  be  cents»  mills,  itnd  parts  of  a  mill. 

Jn  the  same  zcay  is  calculated  the  interest  on  any  sum  of  money  in  pounds,  shil- 
lings, pence  and  farthings,  with  this  difference  only,  that  the  tzco  figures  cut. 
off"  to  the  right  hand  of  pounds,  must  he  reduced  to  the  loxjDest  dciiomination, 
each  time  cutting  off  as  at  first. 

2.  For  tivo  or  more  years,  multiply  the  interest  of  one  year  by  the  num- 
ber of  years. 

3.  For  months,  take  proportional  or  aliqiiot  parts  of  the  interest  for  one 
yestr,  that  is,  for  G  months,  i ;  for  4  months,  -J-  ;  for  3  months,  -]-,  &.c. 

For  days,  the  proportional  or  aUquot  parts  of  the  interest  for  one  month, 
allowing  30  days  to  a  month. 

EXAMPLES. 
1.  What  is  the  interest  of^86,44C  for  one  year,  at  6 per  cent? 

OPERATION. 

Dolls,  cts.  mills.  In  the  product  of  the  principal  mul- 

86      44      6  principal.  tiplied  by  the  rate  is  found  the  answer. 

G  rate.  Thus  cutting  oif  the  two  right  hand 

figures  from  the  dollars  leave  five  on 

5J    18       67      6  interest.  the   left  hand   which  is  dollars  ;   the 

two  figures  cut  oil  (18)  arc  cents,  th« 
aext  figure  (6)  is  mills  ;  all  the  figures  which  may  cliaace  to  be  at  the  right 
kand  of  mills,  are  parts  of  a  mill ;  hence  we  collect  the  Ans.  ^5  IQcts.  6,^^^^;. 

•  III  New- York  the  law  allows  7  per  ceut. 


96  tJlMPLE  INTEREST.  Sect.  II.  6. 

2.  Whut  is  the  interest  of  ^3G5  lids.  Gmills,  for  three  years  7  months 
and  C  da^s  ? 

OPERAflON. 

3  6  5,   1   4  0  principal. 
6  rate. 


•6  months  i)2   1  j  9  0,  8  7  6  inlerest  for  1  year. 
J    3 


G  5,  7  2    6  2  8  interest  for  3  years. 
1  month  ^)  1   0,  9  5    4  3  8  interest  for  (J  motiths. 
6  days  4)      1,82573  interest  for  1  month. 
,36    514  interest  for  6  days. 

^7  8,  0  7  15  3  interest  for  3  years,  7  months  and  6 
days  ;  that  is  $78  87^.   l^^^mills. 

Because  7  months  are  not  an  even  part  of  a  year,  take  two  such  num- 
bers as  are  even  parts,  and  which  added  together  will  make  7  (6  and  1)  6 
months  are  ^  of  a  year,  tlierefore  for  6  months,  divide  the  interest  of  one 
year  by  2  ;  again  1  month  is  ■}  of  6  months,  therefore  for  1  month,  divide 
ihe  interest  of  G  months  by  G.  For  the  days,  because  6  days  are  j  of  a 
mouth,  or  of  30  days,  therefore /or  G  days,  divide  the  interest  of  1  month 
by  5.  Lastly  add  the  interest  of  all  the  parts  of  the  time  together,  the  sum 
is  the  answer. 

3.  What  is  the  interest  of  £71  7s.  GJcZ.  4.  What  is  the  interest  of 
for  1  year  at  6  per  cent  ?  lOs.  8d.  for  1  year  ?       Ans.  1$. 

OPERATION. 

£.     s.     d.     g. 


71     7      6 


6 


y.3'52  .ins.  £4  5j.  7frf. 


Sect.  li.  5.  SIMPLE  INTEREST.  97 

When  the  rate  is  at  6  per  cent,  there  is  not  perliapa  a  more  concise  and 
easy  way  of  casting  interest,  on  any  sem  ol"  money  in  Dollars,  Cents,  and 
Wills,  than  by  the  iollowing 

METHOD. 

Write  down  half  the  greatest  even  number  of  months  for  a  multiplier;  it 
there  be  an  odd  month  it  niiist  be  reckoned  30  days,  for  which  and  the 
given  days,  if  any,  seek  how  many  times  you  can  have  six  in  the  sura  of 
them,  place  the  ligiire  for  a  decimal  at  the  riglit  hand  of  half  the  even  nuiii- 
ber  of  months,  already  found,  by  which  multiply  the  principal  ;  observing 
in  pointing  ofl'  the, product  to  remove  the»decimal  point  or  separatrix  tzio 
figures  from  its  natural  place  towards  the  left  hand,  that  is,  point  off  txro 
more  places  for  decimals  in  the  product,  than  there  are  decimal  places  in 
the  multiplicand  and  multiplier  counted  together ;  then  all  the  ligurcs  to 
the  left  hand  of  the  point  will  be  dollars,  and  those  to  the  righfhand,  dimes, 
cents  and  mills,  &c.  which  will  be  the  interest  required. 

Should  there  be  a  remainder  in  taking  ohg  sixth  of  the  days,  rediKe  it  to 
a  vulgar  fraction,  for  which  take  aliquot  parts  of  the  multiplicand.    Thus. 

If  the  remainder  be  1=|,  divide  the  multiplicand  by  G 
If 2=i, by  3 

If 3=^, -    by  2 

If 4=|, by  3  twice. 

If b=\,  and  i, by  2  and  3. 

The  quotients  which  in  this  way  occur,  must  be  added  to  the  product  of 
the  principal  multiplied  by  half  the  months,  Lc  the  sum  thus  produced  will 
be  the  interest  required. 

IMien  there  are  days,  but  a  less  number  than  G,  so  that  6  cannot  be  contain- 
ed in  them,  put  a  cypher  in  place  of  the  decimal  at  the  right  hand  of  the 
months,  then  proceed  in  all  respects  as  above  directed. 

Note.     In  casting  interest,  each  month  is  reckoned  30  days. 

EXAMPLES. 
1.  What  is  the  interest  of  ^76,54  for  1  year,  7  months  and  11  days  ? 

OFERATION. 

The  number  of  months  being  19,  the  greatest 
even  number  is  18,  half  of  which  is  9,  which  I 
write  down  ;  then  seeking  how  often  6  is  con- 
tained in  41,  (the  sum  of  the  days  in  the  odd 
month  and  given  days)  1  find  it  will  be  Gtime?. 
which  I  set  down  at  the  right  hand  of  half  th'- 
even  number  of  months  for  a  decimal,  by  whi^h 
together  1  multiply  the  principal.  In  laki-y- 
Ans.  7,  4  1  1  6  2  one  sixth  of  the  days  (41)  there  will  bev<i  re 
^^^"v^^  mainder  of  6=]-  a,id  J  for  which  I   take,  ihit 

ti     ^     ri  one  half  the  multiplicr.iid,   that   is.  divide  ikt 

'^     g    ^  multiplicand  by  2,  then  by  3,  and  these  qu#- 

^     '^     ^  tients  add»!d,  with  the  products  of  half  the  even 

number  of  months,  &.c.  the  sum  cf  thrMii  will 
shew  the  interest  required,  observing  to  count  c(l  tjio  more  figures  for  de- 
cimals in  the  product  than  there  arc  decimal  figures  in  both  the  nr.uhiplter 
and  multiplicand  counted  togetlior. 

For  the  concis.iooss  and  simplicity  cf  the  above  method  it  is  coT?ceivcd 
tliat  instructors  wi!'  :-o:ouin):T.:l  ii  to  their  pupils  in  praftrcncc  to  ?.ny  other 

N 


7  6, 

5 

4 

9, 

6 

9  6  9 

2 

4 

6 

8  8  8 

6 

t 

3  8 

2 

7 

2  5 

5 

7 

98 


SIMPLE  INTEREST. 


Sect.  II.  S. 


2.  What  is  the  interest  of  g6,93 
for  2  years  and  8  months  ? 

^n«.  diets.  6m. 


3.  What  is  the  interest  of  $61,69 
for  3  years  and  2  months  ? 

^nj.  J 12  84<:/».  7m, 


4.  What  is  the  interest  of  91 
cents  for  27  years  ? 

Ji7is.  $1  41cts.  4m. 


When  the  interest  on  any  sum  is 
required  for  a  great  number  of 
years  it  will  be  easier  first  to  find 
the  interest  for  1  year,  then  mul- 
tiply the  interest  so  found  by  the 
number  of  years.. 

6.  What  is  the  interest  of  ^2870,32 
for  10  days  ?    Jlns.  $4  IBcts.  3m. 


When  the  rate  is  any  other  than  6  per  cent,  first  find  the  interest  at  6 
per  cent,  then  divide  the  interest  so  found  by  such  parts  as  the  interest  at 
the  rate  required  exceeds  or  falls  short  of  the  interest  at  6  per  cent,  and 
tlie  quotient  added  to  or  subtracted  from  the  interest  at  6  per  cent,  as  the 
c^vo  iiray  he,  will  give  the  interest  at  the  rate  required.  -f^  ^'^^^ 

G.  What  i(5  the  interest  of  $137,  84     7.   What  is  the  interest  of  ^Br)7 
(c)r  2  years  and  6  months,  at  6  oer        for  10  months  at  8  per  cent?  ^ 
cent?  Jins.pl, 23.  Ans.  $5,^11. 


Sect.  H.  5. 


SIMPLE  INTEREST. 


d9 


8.  What  is  the  interest  of  $^,29 
for  1  month  19  days  at  3  per  cent  ? 
j3mj.  9  mills. 

10.  What  is  the  interest  of  ^1600 
for  1  year  and  3  months  ? 

Jlns.  ^120. 

^2.  What  is  the  interest  of  $11,68 
for  11  months  and  28  days  ? 

Alls.  ^1,054. 

14.  What  is  the  interest  of  gl05 
,61  for  1  year  7  months  and  6  days  ? 
Ans.  glO   I3cts.  8m. 


9.    What  is  the  interest  of  $18 
for  2  years  14  days,  at  7  per  cent  ? 
J?is.  g2  b6cis.  9m. 

11.  What  is  the  interest  of  ^6,811 
for  1  year  and  1 1  months  ? 

Ans.  GGcts.  8m. 

13.  What  is  the  interest  of 
^861,12  for  9  months  25  days,  at 
7  per  cent?  Ans.  $-19,394. 

15.  What  is  the  interest  of  $86 
for  9  months  ?  Ant.  $3,81. 


16-  What  is  the  interest  of  ^78,36  17.  What  is  the  interest  of  ^612 
for  5  years  10  months  and  3  days  ?  30  cents  foi*«  years  8  months  and 

Ans.  $^1  46cts.  5m.  4  dai's  .'  .4ns.  $130,509. 

To  this  mode  of  computing  interest,  I  would  add  from  the  "  Massachusetts 
Justicc^^  a 

METHOD 

Of  computing  the  interest  due  upon  Bonds,  Notes,  ^c.  when  partial  payments 
may  at  di^'erent  times  be  made,  as  established  by  the  Courts  of  Law  in  Mas- 
sachusetts. 

RULE. 
Cast  the  interest  up  to  the  first  payment,  and  if  the  paytoent  exceed  the 
interest,  deduct  the  excess  from  the  principal,  and  cast  the  interest  upon 
the  remainder  to  the  time  of  the  second  payment.  If  the  payment  be  less 
than  the  interest,  place  it  by  itself,  and  cast  on  the  interest  to  the  time  of 
the  next  payment,  and  so  on  until  the  payments  exceed  the  interest,  then 
deduct  the  excess  from  the  principal  and  proceed  as  before. 

EXAMPLES. 

Suppose  A  should  have  a  bond  against  B  for  1166  dollars  66  cents  and 
6  mills,  dated  May  1,  1796,  upon  which  the  following  payments  should  be 
made,  viz : 

Dolts.  Mills. 

1.  December  2-5,  1796 166,666 

2.  July  10,  1797 16,666 

3.  September  1,  1798 50,000 

4.  June  14,  179e 333,333 

5.  April  15,  1800 620,000 

What  will  be  due  upon  it  August  3,  1801  ? 

To  facilitate  the  operation,  l«t  the  spaoo  of  time  from  tl^  date  of  the 
Eond  to  the  day  of  the  liist  payroont,  and  from  the  time  of  oti*  payment  to 
thst  of  another,  and  from  that  of  the  last  payment  to  the  time  of  s^Hticroent, 
■■•■*'  Si,-*-  coiuiM'tod   and  sf-t  ijM.vn  against  tbe  da|FoV  paymenc  ■<i^  above.— 

\ 


Months. 

Days. 

7 

24 

6 

15 

13 

21 

9 

13 

10 

1 

15 

18 

IS.  ^231 

,76. 

100 


SIMPLE  INTEREST 


Sect.  II.  5. 


Then  set  down  the  sum  on  which  the  interest  is  to  be  cast,  with  the  interest 
and  paymonts  in  columns  thus. 


h'nneipal.    |       Time.      |     Interest.    (  Payments.  |       Excess. 


UuHs.  Mils 

1166,666 

121,167 


1045,499 
1045,499 
1045,490 

245,093 


800,406 
579,847 


220,559 


Mo.  Da. 
7  24 


6 

13 

9 


10 


15 
21 
13 


U 


13 


Dolls.  M. 
45,499 


33,978 
71,616 
49,312 


154,906 
40,163 

17,203 


Dolls.  M. 
166,666 


16,666 

50,000 

333,333 


The  last  remainder 

Interest  from  the  last  payment 

Sum  due  August  3,  1801 


399,999 
620,000 


220,559 
17,203 

237,762 


Dolls.  M. 
121,167 


245,093 
579,847 


\ 


2.  Supposing  a  note  of  867  dollars  33  cents,  dated  January  6, 1794,  upon 
whifth  the  following  payments  should  be  made,  viz. 

1.  April  16,  1797  gl36,44d«. 

2.  April  16,  1799  319, 

3.  Jan.      1,  1800  618,68 
What  would  be  due  July  11,  1801  ?                   Ans.  ^215,103. 

r 


I 


Sect.  II.  6.  SUPPLEMENT  TO  S.  INTEREST.  10. 

SUPPLEMENT  TO  SIMPLE  INTEREST. 


1. 

2. 
3. 

4. 
6. 
6. 
7. 
8. 
9. 

10. 

11. 
12. 


QUESTIONS. 
What  is  Interest  ? 

What  is  understood  by  6  per  cent  ?  3  per  cent  ?  8  per  cent,  &c 
What  per  cent  per  annum  is  allowed  bylaw  to  the  lender  for  the  use 

of  his  money  ? 
What  is  understood  by  the  principal  ?  the  rate  ?  the  amount  ? 
Of  how  many  kinds  is  interest  ?  in  what  does  the  difference  consist 
How  is  simple  interest  calculated  for  one  year  in  Federal  Money  ? 
For  more  years  than  one,  how  is  the  interest  found  ? 
When  there  are  months  and  days,  what  is  the  method  of  procedure  ' 
What  other  method  is  there  of  casting  interest  on  sums  in  Federa* 

Money  ? 
When  the  days  are  a  less  number  than  6,  so  that  6  cannot  be  con- 
tained in  them,  what  is  to  be  done  ? 
How  is  simple  interest  cast  in  pounds,  shillings,  pence  and  farthings  r 
When  partial  payments  are  made  at  different  times,  how  is  the  in* 
terest  calculated  ? 


EXERCISES, 

1.  What  is  the  interest  of  ^916,72 
for  1  year  and  4  months  ? 

Ans.  $73,337. 


2.  What  is  the  interest  of  $93, 
nets,  for  11  days? 

Ant.  n  centi. 


3.  What  is  the  interest  of  $5,19 
for  7  months  ?       .Iny.  \8cts.   Im. 


I 


4.  What  is  the  interest^of  $1,07 
r  3  years,  G  months  and  16  days  ? 
Jlna.  22cts.  Im. 


102 


SUPPLEMENT  TO  S.  INTEREST. 


Sect.  II.  6. 


6.  What  is  the  interest  of  two  6.  What  is  the  interest  of  nine 
hundred  dollars  and  six  cents,  4  cents  46  years  7  months  and  11 
days  ?  An$.  IScta.  3m.         days  ?  Ans.  24c tt.  Gfn. 


7.  What  is  the  interest  of  half  a 
mill  667  years  ?        Ans.  let.  Im. 


8.  A's  note  of  ^365,37  was  given 
Dec.  3^  1797  ;  June  7,  1800  he  paid 
^97,16  ;  what  was  there  due  Sept. 
11,1800  2        ^«s.g328,32. 


9.  B'9  note  of  ^175  was  given 
Dec.  6,  1798,  on  which  was  endor- 
sed one  year's  interest :  what  was 
there  due  Jan.  1,  1803  ? 

An$.  ^207,22. 


10.  C's  note  of  ^56,75  was  given 
June  6,  1801,  on  interest  after  90 
days;  what  was  there  due  Feb.  9, 
18Q2?  j2im.  ^68,19. 


1 1 .  D's  note  of  two  hundred  three 
dollars  and  seventeen  cents  was 
given.Oct.  6,  1808,  on  interest  after 
3  months  ;  Jan.  5,  1809,  he  paid  fifty 
dollars ;  what  was  there  due  May  2d, 
1811?  ^ns.  ^174,  53. 


12.  E's  note  of  eight  hundred 
seventy  dollars  and  five  cents,  was 
given  Nov.  17,  1800  on  interest  after 
90  days  ;  Feb.  11,  1805  he  paid  one 
hundred  eighty  six  dollars  and  six 
cents  ;  what  was  there  due  Dec.  23, 
1807?  .^n*.  $J045,  34. 


SicT.  II.  5.  SUPPLEMENT  TO  S.  INTEREST.  103 

13.  What  is  the  interest  of  £41  lis.  Sid.         14.  What  is  the  interest  of 
for  a  y«ar  and  2  months  ?  ^273,61,  at  7  per  cent  for  1 

Jins.  £2  18s.  2^cL  year  and  10  days  ? 

Ms.  g;  19,677. 


15.  Supposing  a  note  of*  ^317,92,  dated  July  5,  1797,  on  which  were 
the  following  payments— Sept.  13,  1799,  ^208,04;  March  10,  1800,  $1G^ 
what  was  the  SHm  due  Jan.  1,  1801 1  Ana.  ^83,991. 


104 


COMf'OUXD  INTEREST. 


Sect.  II.  5 


C031P0UND  INTEREST 

Is  calculated  by  arlding  the  Interest  to  the  principal  at  the  end  of  each 
year,  and  making  the  anjount  tlie  principal  lor  the  succeeding  year  ;  then 
the  given  principal  subtracted  iVoni  the  last  amount,  the  remainder  will  be 
the  compound  intereit. 

A  concise  end  easy  method  of  casting  Compound  Interest,  at  6  per  cent  on  any 
sum  in  Federal  .Money. 

RULE. 

Multiply  the  given  sum,  if 

For  2  years  by  112,36  For  7  years  by  150,3630 

3 years  —  119,1016  8  years—  159,3848 

4  years  —  126,2476  9  years  —  168,9478 

5  years  —  133,0225  10  years  —  179,0847 

6  years—  141,8519  11  years—  189,0298 

Note  1.  Three  of  the  first  highest  decimals  in  the  above  numbers  will 
be  sufnciently  accurate  for  most  operations  ;  the  product  remembering  to 
remove  the  separatrix  ttvo  figures  from  its  natural  place  towards  the  left 
lno'l.  will  then  shew  the  amount  X)f  princii)al  and  compound  interest  for 
Ilic  given  number  cf  years.  Subtract  the  principal  from  the  amount  and  it 
will  sliew  the  compound  interest. 

2.  When  there  are  months  and  days  ;  first  find  the  amount  of  principal  and 
compouml  interest  far  the  years,  agreeable  to  the  foregoing  method,  then 
for  the  months  and  days  cast  the  simple  interest  on  the  amount  thus  found  ; 
this  added  to  the  amount  will  give  the  answer. 

3.  Any  surn  cf  money  at  Compound  Interest,  will  double  itself  in  11 
years  10  months  and  22  days.  » 

EXAMPLES. 
1.  What  is  tlie  compound  interest         2.  What  is  the  amount  of  ^236  at 
of  ^56,75  for  11  years  ?  compound  interest  for   4  years,  7 

months  and  six  days  ? 


OPERATION. 

5  6,  7  5 

1  8  9, 8    2  9 

^ 

5  10    7  5 
113  5    0        « 
4  5  4  0  0 
6   10  7  5 

4  5  4  0  0 

5  6  7  5 


OPERATION. 

1  2  G,  2  4  7  6 
2  3  6 

7  5  7  4  8  5  6 
^787426 
2  5  2  4  9  5  2 

^2  9  7,  9  4  4  3  3  G  Amount/or  4 yrt, 
3,  6 


•   I    0  7,72795    75  Amount.      1787664 
5  6,  7  5  principal  subtracted.        0  9  3  8  3  2 


Ip  5  0,  9  1  conij)ound  interest.     ^1   0,  7  2  5  9  8  4intcrcstfor7  mo.G  days. 

2  9  7,  9  ,4  4  amount  for  4  years  added. 


$3  0  8,  6  6  9 


Sect.  II.  6.  COMPOUND  MULTIPLICATION.  105 

\  6.  COMPOUND  MULTIPLICATION. 


Compound  Multiplication  is  when  the  Multiphcand  consists  of  several 
denominations.     It  is  particularly  useful  in  finding-  the  value  of  Goods. 

The  different  denominations  in  what  was  formerly  called  Lanful  Money, 
render  this  rule  Avith  some  others  in  Arithmetic,  as  Compound  Division  and 
Practice,  rules  of  great  usefulness,  quite  tedious,  and  the  variety  of  cases 
necessarily  introduced,  extremely  hurthensome  to  the  memory. — This  lum- 
ber of  the  mind  might  be  almost  wholly  dispensed  with,  were  the  habit 
of  reckoning  in  Federal  Moncij  generally  adopted  throughout  the  United 
States. 

For  important  reasons,  pow/jrfs,  shillings,  pence  SiTid  farthings,  ought  to  fall 
wholly  into  disuse  :  Federal  Money  is  our  national  currency  ;  the  scholar 
might  encompass  the  most  useful  rules  in  Arithmetic  in  half  the  time  ;  the 
value  of  commodities  bought  and  sold,  might  be  cast  with  half  the  trouble, 
and  with  much  less  liability  to  errors,  were  all  the  calculations  in  money 
universally  made  in  Dollars,  Cents,  and  AJitls.  But  this,  to  be  practised, 
must  be  taught  ;  it  must  be  taught  in  our  schools,  and  so  long  as  the  prices 
of  goods,  and  almost  every  man's  accounts  are  in  Potinds,  Shillings,  Pence 
and  Farthings,  this  mode  of  reckoning  must  not  be  left  untaught. 

To  comprise  the  greater  usefulness,  and  also  to  shew  the  great  advan- 
tage which  is  gained  by  reckoning  in  Federal  Money,  I  have  contrasted  the 
two  modes  of  account,  and  in  separate  columns  on  the  same  page,  have  put 
the  same  questions  in  Old  Lawl'ul  and  in  Federal  Money. 

OPERATIONS. 


IN  POUNDS,  SHILLINGS,  PENCE,  FARTH. 
C.ISK    I. 

When  the  quantity  does  t}ot  exceed 
12  yards,  pounds,  4'C.  sot  down  the 
price  of  one  yard  or  pound,  and 
J. hoe  the  quantity  underneath  the 
lowest  denomination  for  a  multiplier. 
Begin  by  inultiiilying  the  lowest  do- 
nomination,  and  carry  by  the  same 
ruios  from  on«  denomination  to  an- 
other, as  in  Comjiound  Addition. 

KXAMPLES. 
1.  What  will  7  yards  cf  cloth  coat 
at  9s.  5(/.  jiCT  yard. 

OPERATION. 

£.      *.      d. 

0         9        5  price  of  1  yard.^ 

7  yards. 


Ans.  3         6       11  price  of  1  yards. 

1  say  7  times  5  is  35  pcnce=2«.  1 1(7. 
I  set  down  ]  1  aid  carry  i',  saying  7 
limes  V  is  63,  and  2  I  carry  are  Gi^. 
-~.£3  5s.  which  I  setc^ouD. 


IN    DOLLARS,    CENTS,    MILLS. 

AV  JILL  CISES. 

Multiply  the  price  and  the  quantity 
together,  according  to  the  rules  of 
multiplication  in  Decimal  Fractions, 
and  the  product  will  be  the  answer. 
That  is, 

Multiply  as  in  Simple  Multiplica- 
tion, and  from  the  product  point  oflf 
so  many  places  for  cents,  and  mills, 
as  there  arc  places  of  cents  and  muls 
in  liie  price. 

EXAMPLES. 
].  What  will  7  3  ards  of  cloth  cost 
at  'i',!,!'^!  (equal  to  9s.  St/.)  per  yard  ? 

OPERATION. 

D.     cts.  As  there  are 

I,      57  price.      two     decimal 

7  quantity.  \}hces  in  the 

price,     so     I 


Jlns.  10        99  pn're  rf  make  two  in 
1  yards,    the  product. 


106 


COMPOUND  MULTIPLICATION. 


&BCT.  II.  6. 


POUNDS,  SHirXINSS,  PENCE,  FARTH. 

2.  What  will  9  pounds  of  sugar 
cost  at  \0d.  per  lb.  ? 

Ms.  7s.  Gd. 


3.  What  will  6  yards  of  cloth  cost 
at  £1    10s.  5d.  per  yard  ? 

Ms.  £9  2s.  Gd. 


DOLLARS,    CENTS,    MILLS. 

2.  What  will  9  pounds  of  sugar 
cost  at  go,  139  per  lb.  ?  Ans.  $1,251. 


CASE  2. 

When  the  quantity  exceeds   12,  and 

is  any  number  zmlhin  the  Multiplication 

Table,  multiply  by  two  such  numbers 

as    when  multiplied   together,   will 

reduce  the  given  quantity. 

If  two  numbers  will  not  do  this  ex- 
acily,  multiply  by  two  such  numbers 
J-.*  come  the  nearest  to  it,  and  by  the 
deficiency  or  excess  multiply  the 
multiplicand,  and  this  product  added 
to  or  subtracted  from  the  first  pro- 
duct as  the  case  may  require,  gives 
the  answer. 

EXAMPLES. 

1.  What  v/ill  42  yards  of  cloth 
cost  at  15s.  9 J.  per  yard  ? 

OPERATION. 

£.  s.  d. 

0    15  9  price  of  1  yd. 
Multiplied  by  6 


Multiplied  by 


14  6  price  of  G  yds. 

7 


Ans.  33      1    6ptnceofA2yds. 
Beca'ise  6  times  7  is  42, 1  multiply 
the  price  of  1  yard  by  6,  and  this 
product  by  7,  as  the  rule  directs. 


3.  What  will  6  yards  of  cloth  co^t 
at  ^5,07  per  yard  ? 

Am.  $30,42. 


at  $2 


What  will  42  yards  of  cloth  cost 
,625  per  yard  ? 

OPERATION. 

D.    cts.    m. 
2,     6  2     5 
4     2 


5 
1  0  5 


2  5 
00 


Ans.  110      2  5    0 


1 


SecT.  II.  6. 


COMPOUND  MULTIPLICATION. 


107 


POUNDS,  SHILLINGS,  PENCE,  FARTH. 

2.  What  will  125  yards  of  cloth 
Cost  at  5s.  Id.  per  yard  ? 

Ms.  £34   17s.   llrf. 


3.  What  will  61  pounds  of  tea  cost 
at  3*.  6d.  per  lb.  ? 

Ans.  £8  18s.  6d 


4.  What  will  130  yards  of  cloth 
cost  at  £2  3s,  9d.  per  yard  ? 

Ans.  £284  7s.  6d. 


CASE  3. 

When  the  multiplier,  that  is,  the 
ijuantity,  exceeds  144,  multiply  first 
by  10,  and  this  product  again  by 
10,  which  will  give  the  price  of  100 
j'ards,  &c.  and  if  the  quantity  be 
even  hundreds,  multiply  the  price 
of  100  by  the  number  of  hundreds 
in  the  question,  and  the  product 
will  be  the  answer  ;  if  there  be  odd 
numbers,  multiply  the  price  of  10 
by  the  number  of  tens,  and  the  price 
of  unity,  or  1 ,  by  the  number  of  units, 
♦hen  these  several  products  added 
together  will  be  the  answer. 


DOLLARS,    CENTS,    MILLS. 

6.  What  will  125  yards  of  cloth 
cost  at  93  cents  per  yard  ? 

Ans.  gl  16,25. 


6.  What  will  51  pounds  of  tea  co«t 
at  ^0,583  per  lb.  ? 

Ans.  ^29,733. 


7.  What  will  130  yards  of  cloUi 
cost  at  ^7,26  per  yard  ? 

Ans.  |942,50. 


'^^ 


1U3 


COMPOUxVD  MULTIPLICATION. 


Sect.  IL  &  ' 


POUNDS,  SHILI.I^GS,  PENCE,  FARTH. 

EXAMPLES. 
1.  What  Trill  563  yanJs  of  clotli 
cost  at  £l  6s.  Id.  ])or  yard  ? 

OrEP.AI  ION. 

£.    s.     d. 
1       6       1  price  nj  1  yard. 
10 


13      5    10  price  nf  \0  yds. 
10 


132    18      ^  price  of  100  yds. 


GGl     11       Q  price  of  500  yds. 

Wy^l  <     ''^^     ^'^      0  price  of  60  ?/c?s. 
3  times  1yd.  3    19      9  price  of  3  yds. 

Ans.    7'18      6      r)  price  of  563  yds. 

2.  What  will  328  yards  of  cloth 
cost  at  10^.  6^d.  per  yard  ? 

Ans.  £172   17s.  8d. 


DOLLARS,    CENTS,    MfLLS. 

8.  What  will  563  yards  of  cloth 
cost  at  ^4,43  per  yard  ? 

OPERATION. 

Yds.  5     G     3 
$4,   4     3 


16  8     9 

2     2     6  2 
2     2     5     2 

2     4     9     4,  0    9  Am. 


3.  What  will  624  yards  of  cloth 
co*t  at  125.  Qd.  per  yard  ? 

Ans.  £395  4s. 


9.  What  will  328  yards  of  cloth 
cost  at  ^  1 ,757  per  yard  ? 

A71S.  ^576,296. 


10.  What  will  624  yards  of  clotb 
cost  at  J^2,lll  per  yard  ? 

Ans.  $1317,261. 


Sect.  II.  6,      SUPPLEMENT  TO  C,  MULTIPLICATION. 


109 


SUPPLEMENT  TO  C03IP0UND  MULTIPLI- 
CATION, 


QUESTIONS. 

1.  What  is  Corapouad  Multiplication? 

2.  What  is  its  use  ? 

5.  Are  operations  more  easy  in  Old  Lawful  or  Federal  Money  ? 
4.  What  is  the  Rule  of  Compound  Multiplication  ? 

6.  When  the  quantity,  that  is  the  Multiplier,  exceeds  12,  and  is  within 

the  Multiplication  Table,  what  are  the  steps  to  be  taken  ? 
iB.  When  no  two  numbers   multiplied  together  will  produce  the  given 

quantity,  what  then  is  to  be  done  ? 
^.  When  the  multiplier  exceeds  144,  what  is  the  method  of  procedure  ? 
8.  When  the  price   of  goods  are  given  in  Federal  Money,  what  is  the 
general  and  universal  rule  for  fimling  their  value  by  Multiplication  ? 
EXERCISES. 
1.  A  man  has  38  silver  cups,  each         2.    If  a  man  travel  34  miles,   3 
weighing    loar.    Spxets.    I6grs.    how     furlongs,   and   17  rods  in  one   day, 
much  silver  do  they  all  contain  ?  how  far  will  he  travel  in  62  days  ? 

Ans.  3/6.  &0Z.  19  pwts.  8grs.  Ans.  2134  7m7cs,  4  fur.  14  rodi. 


3.  What  will  235  yards  of  cloth 
come  to  at  £1  2s.  5^1.  per  yard  ? 
4»s.£263  17*.  8^d. 


4.  If  a  horse  nm  a  mile  in  12 
minutes,  16  seconds,  in  what  time 
would  he  go  176  mlies  ? 

Ans.   ID.   llh.  58m.   56*cc. 


110 


COMPOUND  DIVISION.  Sect.  II.  7 

7.  COMPOUND  DIVISION. 


COMPOUND  DIVISION  is  the  dividing  of  different  denominations. 

OPERATIONS. 


IN  POU.VDS,  SHILLINGS,  PENCE,  FARTH. 

CASE  1. 

1.  niun  the  divisor,  that  is,  the 
quantity,  does  not  exceed  12,  begin  at 
the  hip;hesit  denomination,  and  in  tlie 
manner  of  short  division,  find  how 
many  times  tiie  divisor  is  contained 
in  it  ;  place  the  quotient  under  its 
own  denomination,  and  if  any  thing 
remain,  reduce  it  to  the  next  less 
denomination,  and  divide  as  before  ; 
to  proceed  through  all  the  denomi- 
nations. 

2.  Jf  the  qvantity  exceed  12,  and 
there  he  any  two  numbers  nhich  vndti- 
plird  together  rt'ill  produce  it,  divide 
the  price  first  by  one  of  those  niim- 
beis,  and  this  quotient  by  the  other. 

EXAMPLES. 

1.  If  5  yards  of  cloth  cost  £3  13s. 
Sd,  what  is  that  per  yard  ? 

Ol-ERATION. 

£.  s.  d. 
6)3   13  6  price  of  5  yards. 

0  14  Q^  price  of  1  yard. 

Finding  I  cannot  have  the  divisor 
(5)  in  the  first  denomination  (£3)  I 
reduce  it  to  shiUings,  (60)  and  add 
in  the  13  shillings,  which  make  73 
.^hiIlings  in  which  the  divisor  (5)  is 
confined  14  times  and  3  remain  ;  I 
set  dof'n  the  14,  and  the  remainder 
(3  shillings)  reduce,  to  pence  (36) 
niid  the  Gd.  added  make  42  pence  in 
which  the  divisor  is  contained  eight 
times  and  icw>  remain  ;  I  set  down 
ti.e  8  and  reduce  the  2  pence  to 
f*rthings  (8)  in  which  I  have  the  di- 
visor once  (1  qr.  or  ^u'.)  and  aremain- 
dei-  of  ^  of  a  farthing,  which  being  of 
small  value  is  neglected. 

2.  If  48  y^r'U  of  clolh  cost  £4 
ICs.  4-^ii.  vvhut  j3  that  per  y.ird  ? 

Jiij.'£o  2s. 


IN    DOLLARS,    CENT*,    MILLS.  1 

IN  ALL  CASES. 
Divide  the  price  by  the  quantity^ 
and  point  off  so  many  places  for 
cents  and  mills  in  the  product  ag 
there  are  places  of  cents,  and  mills 
in  the  dividend. 

If  the  quantity  he  a  composite  num- 
ber, that  is  produced  by  the  multipli- 
cation of  two  numbers,  the  operation 
may  be  varied  by  dividing  the  price 
first  by  one  of  those  numbers,  aod 
this  quotient  by  the  other. 

EXAMPLES. 
1.  If  5  yards  of  cloth  cost  g  12,25, 
what  is  that  per  yard  ? 


D. 

5)12, 


OPERATION. 

Cts. 
25 


Jins.  2,     46 


There  are  two 
decimal  places  ia 


the  dividend.     I 
therefore    point 

•ff  two  places  for  decimals  or  cents 

in  the  quotient. 


2.  If  48  yards  of  cloth  cost  $16,06, 
what  is  that  per  yard  ? 

Ans.  $0  33  cents* 


Sect.  II.  7. 


COMPOUND  DIVISION. 


Ill 


PO'   .t>S,  SHI.    .,"  ^i'GS,  PENCE,  FART   ',. 

3.  If  24^6.  of  tea  cost  £2  75.  9|c/. 
srhat  is  that  per  /6.  ? 

Ans.  £0  Is.  ll|rf. 


4.  If  36  yards  of  cloth  cost  £42 
§1.  71c?.  what  is  that  per  yard  ? 
An$.  £1  4s.  2i(/. 


CASE  2. 

1.  "  Having  the  price  of  a  hundred 
weight  (1 12/i.)  to  find  the  price  of  Mb. 
divide  the  given  price  by  8,  that 
quotient  by  7,  and  this  quotient  by  2, 
and  the  last  quotient  will  be  the  price 
©f  1/6.  required." 

2.  If  the  number  of  hundred  u-eight 
be  more  than  one,  first  divide  the 
whole  price  by  the  number  of  hun- 
dreds, then  proceed  as  before. 

EXAMPLES. 
1.  \i\cTit.  of  sugar  cost  £3  ".s.  &d. 
what  is  that  per  lb.  ? 

OPERATION. 

£.  s.  d.  q. 

8)3    7    6      price  of  \cwt. 

7)0    8     5     1  price  of  Ulh.  or  ic^L 
2)       1     2    2priceof2lb.  or-'^czvi.    j 
.'?«s.    0     7      1  priiy-  »/  !  '•■. 


DOLLARS,    CENTS,    MILLS. 

3.  If  24/6.  of  tea  cost  $7,97  what 
Is  that  per  lb.  ? 

Am.  $0,233. 


4.  If  35  yards  of  cloth  cost  $141, 
103,  what  is  that  per  yard  ? 

Ans.  $4,031. 


The  same  may  be  done  in  Federal 
Money. 


6.  If  1  cwt.  of  sugar  cost  $11,25, 
what  is  that  per  lb.  '! 

Ans.  iO  cents. 


112 


C031P0UND  DIVISION. 


Sect.  il.  7. 


rOUVDS,  SHILLINGS,  PE.VCE,  FARTII. 

2.   If  8ca/.  of  cocoa  cost  £15  7s. 
4d.  what  is  that  per  lb.  ?     Jhis.  4d. 


nOIXARS,    CENT6,    MILLS. 

6.  Uilcxct.  of  cocoa  cost  g51,223, 
what  i«  tl)at  per  lb.  ? 

Ant.  bets.  7m. 


3.  If  5fr:'<.  of  sugar  cost  £15  135. 
what  is  that  per  lb.  ? 

Jlns.   lid. 


CASE  S. 

"  When  the  divisor  is  such  a  number 
as  cannot  be  prodvred  by  the  inultipH' 
cation  of  small  nuvibers,  divide  after 
the  manner  of  Ion?  fii>.'i;si(»n,  settins: 
down  the  work  uf  di\idinj^  and  rc- 
ducinff." 


7.  If  Scwt.  of  sugar  cost  ^52^67 
what  is  that  per  lb.  ? 

Ans.  \5cU.  5m. 


Sect.  II.  7. 


COMPOUND  DIVISION. 


IIS 


POUNDS,  SHILLINGS,  PENCE,  FART». 

EXAMPLES. 
1.  If  46  yards  of  cloth  cost  £53, 
lOs.  6u'.  what  is  that  per  yard  .' 

OPERATION. 

£.     s.     d.  £.     s.    d. 

4G)53     10    6(1        3     3i.3ny. 
46 


46)130(3 
138 

12 
12 

46)150(3 
138 

12 
4 

46)48(1 
46 


2.  If  263  bushels  of  wheat  cost 
£86  Is.  lOd.  what  is  that  per  feushel  ? 
Ans.  6«.  e^d. 


3.  If  670  gallons  of  wine  cost 
£147  1».  lid.  what  is  that  per  gal- 
lon ?  Ans.  4s.  4\d. 


DOLLARS,   CENTS,   MILLS. 

8.  If  46  yards  of  cloth  cost  $119-, 
416,  what  is  that  per  yard  ? 

Anj.  ^3,878. 


9,    If  263  bushels  of  wheat  cost 
1^287,973,  what  is  that  per  bushel  ? 
,   Ans.  $1,094. 


10.    If  670  gallons  of  wine  cost 
§490,32,  what  is  that  per  gallon  ? 
Ant.  $0,73. 


114  SUPPLOIENT  TO  COMPOUND  DIVISION.      Sect.  II.  7. 

SUPPLEMENT  TO  C03IP0UND  DIVISION. 


QUESTIONS. 
What  is  Compound  Division  ? 
When  the  price  of  any  quantity  not  exceeding  12,  of  yards,  pounds, 

&.C.  is  given  in  pounds,  shillings,  pence  and  farthings,  how  is  the 

price  of  one  yard  found  ? 
When  the  quantity  is  such  a  number  as  cannot  be  produced  by  the 

multiplication  of  small  numbers,  what  is  the  method  of  procedure  ? 
Having  the  price  of  an  hundred  weight  given,  in  what  way  is  found 

the  price  of  1  lb.  ? 
If  there  be  several  hundred  weight,  what  are  the  steps  of  operating? 
When  the  price  is  given  in  Federal  Money,  what  is  the  method  of 

operating  ? 

EXERCISES. 


rOLNDS,  SHILLINGS,  PENCE,  FARTH. 

1.  If  10  sheep  cost   £4  5s,  Id. 


what  is  the  price  of  each  ? 


Ans.^s.  6irf. 


2.  If  84  cows  cost  £253  13j.  what 
is  the  price  of  each  ? 

Ans.  £3  Os.  4}J. 


DOLLARS,    CENTS,    MILLS. 

Let  the  Scholar  reduce  the  price 
of  sheep  and  of  the  cows  to  Federal 
Money,  and  perform  the  operations 
in  Dollars,  Cents  and  Mills. 

Price  of  1  sheep  $1,426. 


Price  of  1  cow,  ^10,065. 


- 


Sect.  II.  7.     SUPPLEMENT  TO  COMPOUND  DIVISION. 


115 


3.  If  121  pieces  of  cloth  measure 
2896  yards,  1  qr.  3  na.  what  does 
each  piece  measure  ? 

Ans.  23ijds.  3qr.  3na. 


5.  If  2cwt.  of  rice  cost  £2  11j. 
$ld'  what  is  that  per  lb.  ? 

Ms.  nd. 


4.  If  66  tea-$poong  weigh  2Ib. 
lOoz.  14pwt.  what  is  the  weight  of 
each  ? 

Jlns.  lOpwt.  12^grs. 


6.  At  £2  lis.  6-\d.  for  2cwt.  of 
rice,  what  is  that  in  Federal  Money, 
and  what  is  that  per  lb.  ? 

Price  of  lib.  Sets.  8m. 


7.  If  47  bags  of  indigu  wei^h  j  8-  If  8  liorse?  eat 'J«X)bu9hel?  and 
12cwt.  Iqr.  261b.  4oz.  what  does  1  peck  of  oats  in  1  year,  how  much 
each  weigh?  I  will  each  horse  eat  per  day  ? 

Ansi  Iqr.  lib.  12oz,      I  Ans.  Ipk.  Iqt.  Int.  Z^illt. 


116 


SUPPLEMENT  TO  COMrOL'ND  DIVISION'.     Sect.  II.  7. 


Divide   jC-97  25.  3(^  anaonfi;'  4  men,  6  boys,  and  give  each  man  3  times 
5o  ujuch  as  one  boy ;  what  will  each  man  share,  and  each  boy  ? 

OPEHATIOX. 

The  men  have  triple  .£.     *.     d.         £.     s.     d.     q. 

shares,  therefore  inol-     ri)^dl    2     3         (16     10     1      2=1  hoy's  share. 
U\)\y  the  number  o/aion 
'  ;)  by  3,  and  acid  ihe 
nwmbcr    of  boys,   (G) 
Ijf  a  diviaor. 


tncn.  boijs. 

4  h  a 


12 


IC  ihe  number  of 
crjuul  shares  m 
t!is  :..':c/f.=^ Divisor. 


18 

117 

108 

9 

20 


)1U2(10 
18 

2 
J2 

;27(i 


Ans.  4D     10    4      2=?1  ;nan's  *'iare. 

PROOF. 

£49     10     4     2 
4 


198        1     C     Omcr/«sAar«. 

IC     10     1     2  and 
C 


99       0     9     0  boys'  share. 


£237       2     3     0  added. 


9 
4 

3G 


10.  Divide  £39  12s.  ud.  among  4  men,  6  women,  and  9 boys;  give  each 
rjan  double  to  a  woraan,  each  woman  double  to  a  boy. 

£.     s.     d. 

1        15a  boy^s  share. 
.Ins.  (2       2    10  a  ztoman^s  share. 
4       5      u  c  jrtcji.'*  share. 


II.  8.  SINGLE  RULE  OF  THREE.  IT 

^  8.  SINGLE  HULK  OF  THREE, 


THE  Single  Rule  of  Three,  sometimes  called  the  Rl'le  of  PROPORTtoN 
is  knouTi  by  harinij  three  terms  giren  to  find  the  fourth. 

It  is  o{  tiio  kinds,  Direct  and  Indirect,  or  Imcrse. 

SLYGLE  RULE  OF  THREE  DIRECT. 

The  Singrle  Rule  of  Three  Direct  teaches,  by  having  three  numbers 
given  to  find  a  fourth,  which  shall  bear  the  same  proportion  to  the  third 
that  the  second  docs  to  tlie  first. 

It  is  evident,  that  the  value,  weight  and  measure  of  any  commodity  is 
proporiionate  to  its  quantity,  that  the  amount  of  work,  or  consumption  is 
proportionate  to  the  time  ;  thnt  gain,  loss  and  interest,  when  the  time  13 
tixed,  is  proportionate  to  the  capital  sura  from  which  it  arises  ;  and  that  (he 
elTect  produced  by  any  cause  is  proportioned  to  the  extent  of  that  cause. 

These  are  cases  in  direct  proportion,  and  all  others  may  be  known  to  bs 
so,  when  the  number  sought  inrre^scs  or  diminishes  along  with  the  ternj 
from  which  it  is  derived.       Therefore, 

If  viore  require  more,  or  less  require  less,  the  question  is  always  known 
to  belong  to  llie  Rule  of  Three  Direct. 

Alorc  requiring  more,  is  when  the  third  term  is  greater  than  the  first, 
and  requires  the  fourth  term  to  be  greater  than  the  second. 

Less  requiring  less,  is  when  the  third  term  is  less  than  the  first  and  re- 
quif'cs  the  fourth  term  to  be  less  than  the  second. 

RULE. 

**  1.  State  the  question  by  making  that  number  which  asks  the  question, 
"the  third  term,  or  putting  it  in  the  third  place;  that  which  is  of  the 
"  same  name  or  quality  as  the  demand,  the  fir^t  term,  and  that  which  is  of 
"  the  same  name  or  quality  with  the  answer  required,  the  second  term." 

"  2.  Multiply  the  second  and  third  terms  together,  divide  by  the  first, 
"  and  the  t})ioticpt  will  be  the  answer  to  the  question,  which  (as  also  tl;e 
"  remainder)  will  be  in  th(^  same  denomination  in  which  30J  left  the  second 
"  term,  and  may  be  brought  into  any  other  denomination  required." 

The  chief  difficult}'  that  occurs  in  the  Rule  of  Three,  is  the  right  placing 
of  the  numbers,  or  staling  of  the  question  ;  this  being  accomplished,  there 
is  nothing  to  do,  but  to  mi.itiply  and  divide,  and  the  Avork  is  done. 

To  this  end  the  nature  of  every  question  must  be  considered,  and  the 
circumstances  on  which  the  proportion  depends,  observed,  and  common 
sense  will  direct  this  if  the  terms  of  the  qucstioo  be  understooij. 

The  method  of  proof  is  by  inverlieg  the  order  of  the  question. 

»Voie  I.  If  (lie  firct  and  third  terms,  both  or  either,  he  of  different  de- 
nominations, both  terms  muct  be  reduced  to  tlie  lowest  denomination  raen- 
Uoned  in  either,  before  stating  Ihe  que-tion, 

2.  If  the  second  term  consists  of  difi'erent  derjominafions,  it  must  be  re- 
duced to  the  lowest  denomination  ;  tho.  fourth  term  or  nnswer  vi'ill  t'len  be 
found  in  tho  same  denomination,  and  iau?t  he  reduced  back  ?gaiu  to  the 
hig!;esl  denominaiion  pov^ble. 

3.  After  division  if  there  be  any  rcmaind?r,  and  the  qj?oticnt  he  not  in 
tlie  loivest  denomination,  it  must  be  reduced  to  the  ne;;t  less  der.omiiution, 
dividing  ?"  beiore.  So  continue  to  do  till  it  is  bio;;^'iit  to  the  lowest  de- 
nomiPiatipn,  or  till  nothing  remains. 

4.  In  every  qu<"'sti<)n  there  is  a  si^pposilion  and  a  der.'iand  ;  ihc  supposition 
ii  implied  in  ihe  t'.vo  lir:-!  toruisorihc  slalemont,  ihc  <lc-nr.ind  in  the  ihi;n. 


118 


SINGLE  RULE  OF  THREE  DIRECT. 


Sect.  II.  8. 


5.  When  any  of  the  ternas  are  given  in  Federal  Money  the  operation  is 
conducted  in  all  respects  as  in  siinple  numbers,  observing  only  to  place 
the  point  or  separatrix  between  dollar?  and  cents,  to  point  off  the  results 
according  to  what  has  been  tanj^ht  already  in  Decimal  Fractions,  Federal 
Monei/,  and  further  illustrated  in  Compound  Division. 

6.  When  any  number  of  barrels,  bales,  or  other  packages,  or  pieces  are 
given,  if  they  be  of  equal  contents,  find  the  contents  of  one  barrel  or  piece, 
A:c.  in  the  lowest  denomination  mentioned,  which  multiply  by  the  number 
of  pieces,  Lc.  the  product  will  be  the  contents  of  the  whole — If  the  pieces 
&:c.  be  of  unequal  contents,  find  the  content  of  each,  add  these  together, 
and  the  sum  of  them  will  be  the  whole  quantity. 

7.  The  term  which  asks  the  question,  or  that  which  implies  the  demand, 
is  generally  known  by  some  of  these  words  going  before  it ;  How  much  ? 
How  many  ?  How  long  ?  What  cost  ?  What  will  ?  &c. 

EXMIPLES. 
1.  U9lbf.  of  tobacco  cost  65.  what  will  25 /6s.  cost? 


lbs. 
As  9 


lbs. 

25  to  the  answer. 


OPERATION. 

6 

25 

30 
12 

s.   d. 

9)160(16  B  answer. 
9 


Here  25lbs.  which  asks  the  ques- 
tion, (zt'hnt  will  Iblbs.  «S'C.)  is  made 
the  third  term,  by  being  put  in  the 
third  place  ;  dibs,  being  of  the  same 
name,  the  first  term,  and  6s.  of  the 
same  name  with  the  term  sought,  the 
eecond  term. 

I  multiply  the  second  and  third 
terms  together,  and  divide  by  the 
first.  The  remainder  (6)  I  reduce 
to  pence,  and  divide  as  before.  The 
quotients  make  the  answer  I65.  8d. 


60 

54 

6 

12 

9)72(8 
72 

"oo" 

By  inverting  the  order  of  the  question  it  will  stand  thus, 
2.  If  6s.  buy  dibs,  of  tobacco,  what  will  16s.  8d.  buy  ? 


s. 

6 

12 


s. 
16 
12 


72  pence.    200  pence. 


pence. 

As  72 


lbs.     pence. 
9  :  :  200 
200 


Here  the  term  which  asks  the 
question  (16s.  8(Z.)  is  of  different  de- 
nominations ;  it  must,  therefore,  1)6 
reduced  to  the  lowest  denomination 
mentioned  (pence)  as  must  also  the 
other  term  of  the  same  name,  conse- 
quently, to  be  the  first  term. 


72)1800(25/6.  ansa-er. 
144 


360 
360 


Sect.  II.  8.        SINGLE  RULE  OF  THREE  DIRECT.  U9 

.9gain — By  inverting  the  order  of  the  question. 
3.  If  16s.  8d.  (=200  pence)  buy  2oIbs.  of  tobacco,  how  much  will  6j. 
(=72.pc7ice)  buy  ? 

OPER\TION. 

d.         lbs.         d. 
As  200    :    25  :  :  72 

72 

60 
175 


2|00)18|00(9/6j.  j3n».  These  three  questions  arc  only 

18  the  first  varied;  they  shew  how  any 

question  in  this  rula  may  be  inverted. 

4.  If  lor.  of  silver  cost  Gs.  9d.  what  will  be  the  price  of  a  silver  cup 
that  weighs  9oz.  4pwt.  16grs.  ? 

Note. — As  each  of  the  terms  contain 
different  denominations,  they  must  all 
be  reduced  to  the  lowest  denomination 
mentioned. 

Ans.  747  pence,  3^q.  which  must  be 
reduced  to  the  highest  denomi- 
nation, thus, 
pence. 
12)747  Rem.  3d. 

20)62  Rem.  2s. 

£3  2s.  3d.  3|y.  Jint. 


130  SINGLE  RULE  OF  TlIllEE  DIRECT.        Sz(ir.  II.  8. 

6.  If  6  horses  eat  21  bushels  of  oats  in  3  weeks,  how  many  buphcls  will 
20  horses  eat  in  the  same  time  ?  Ans.  10  butheh. 

Tlie  same  qnesiinn  inverted. 
6.  If  20  horses  eat  70  bushels  of 
oats  in  3  weeks,  how  many  bushels 
will  C  horses  eat  in  the  same  time  ? 
Ans.  21  bushels. 


The  stateaicnt  of  every  question  re- 
quires thought  and  consideration  ; — 
here  are  fovr  numbers  given  in  the 
question  ;  to  know  which  three  are  to 

be  employed  in  the  statement,  there  can  be  no  difficulty  if  tbe  scholar  pro- 
ceed deliberately  and  as  hi?  rule  (>irects — first  consider  which  of  the  oriven 
numbers  it  is  that  asks  the  question  ;  that  determined  on,  put  it  in  the 
third  place,  then  seek  for  another  number  of  the  same  name,  or  kind,  put 
that  in  the  first  place,  the  second  place  must  now  be  occtipied  by  that 
cumber  which  is  of  the  same  name  or  kind  with  the  number  sought;  when 
these  steps  are  cautiously  Ibllowed,  the  scholar  cannot  fail  to  make  his 
statement  right. 

7.  If  an  ingot  of  silver  weigh  3Goz.  8.  A  Goldsmith  sold  a  Tankard 
lOpa.'^.  what  is  it  worth  at  6s.  per  for  £10  12.?.  at  the  rate  of  Bs.Ad. 
ouace  ?  Ans.  £,9  2s.  6d.  per  ounce,  I  demand  t'.je  v/eight 

of  it.  Ans.oSoz.  lopzet. 


9.  If  the  moon  move  ISdesr.  lOmtn. 
35sec.  in  one  day  ;  in  what  time  doQ» 
it  perform  one  revohit-on  ? 

Ans.  ^Idai'^.'^kn-.  A.3inin, 


Sect.  II.  8.         SINGLE  RULE  OF  THREE  DIRECT.  121 

10.  If  a  family  of  10  persons  spend         11.    If  a   fattiliv   of  30  persons 

n  bushels  of  rnalt  in  a  month,  how  spend  9  bushels  of  malt  in  a  month, 

many  bushels  will  serve  them  when  how  many  bushels  will  serve  a  farn- 

there  are  30  in  the  family  ?  ily  of  10  persons  the  same  time  ? 
.Ins.  0  hisltrh.  -'ins.  3  lusheh. 


12.  If  12  acres  3  roods,  produce   78  quarters  3  pecks,  Fa/)w  much  will 
35  acres,  1  rood,  20  poles  produce  ?  Am.  21G  qrs.  Obush'  l^peckf. 


^'- 


122 


SINGLE  RULE  OF  THREE  DIRECT.        Sbct.  II.  8 


13.  If  5  acres,    1  rood  produce  26  quarter?,  2  bushels,  how  many  acres 
will  be  required  to  produce  47  quarters,  4  bushels  ?     Ant.  9  acres,  2  mods. 


14.  If  365  men  consume  75  bar- 
rels of  provisions  in  9  months,  how 
much  will  500  men  consume  in  the 
same  time  ?         Ans.  10244-  barrels. 


Note.  In  the  15th  example,  in 
order  to  embrace  the  fraction  (^f 
«/  g.  barrel)  the  integers  102  bar- 
^rels  'must  be  multiplied  by  the  de- 
nominator of  the  fraction  (73)  and 
the  numerator,  (54)  added  to  the 
product. 

After  division,  the  quotient  must 
be  divided  by  the  denominator  of 
the  fraction,  and  (his  last  quotient 
will  be  the  arrswor,  all  which  may 
be  seen  in  the  example. 

The  Scholar  must  remember  to 
do  the  same  in  all  similar  cases. 


15.  If  600  men  consume  102^f. 
barrels  of  provisions  in  9  months, 
how  much  will  365  men  consume  in 
the  same  time  ? 

OPERATION. 

barrels. 

l02^         _    •• 

Multiplied  by       73     the  denominator 

of  the  fraction. 

306 
714 
Add  54  the  numerat&r. 


As  500  :  7600 


::  365 
7500 

182600 
2555 


5j00)27375|00 
73)5476(75  An$. 
511 


365 
366 


16.  How  much  will  4  pieces  of  linen  contriinins:,  vIt:.  35},  56.  37-J-,  and 
.■^8  yards  c»me  to  at  79  cents  j>er  yard  '  Ans.  j^l  16,13. 


Sect.  II.  8.  SINGLE  RULE  OF  THREE  DIRECT.  123 

17.  If  I  give  ^6  for  the  use  of        18.  How  many  tiles  of  8  inches 

^100  for  12  months,    what  must  I  square  will  lay  a  floor  20  feet  long, 

give  for  .'557,82  the  same  time  ?  and  16  feet  broad  ? 

jlns.  $21,469.  .Iiu.  720.  " 


19.  If  2lb.  of  sugar  cost  25  cents,         20.  If  £3  sterling  bo  equal  to  £i 
what  will  100/6.  of  coffee  cost,  if  8/6.     N.    England    currency,    how    much 
of  sugar  are  worth  5/6.  of  coffee  ?         N.  England  currency  will  be  equal 
Ms.  $20.         to  £1000  sterling  ? 

Ans.  £1333  6y.  Sd. 


21.  If  I  buy  7/6.  of  sugar  for  75  N,  B.  Suras  in  Federal  Money 
rents,  how  much  can  I  buy  for  6  are  of  the  s;une  denomination  when 
dollars  ?  Ans.  66/6.         the  decimal  places  in  each  are  equal. 

To  reduce  satins  in  Federal  MoJicy 
to  the  same  denomination,  annex  so 
many  cyphers  to  that  sum  which  h;is 
the  least  number  of  decimid  place?, 
or  places  of  cents,  mills,  ^c.  ;ts  shall 
make  up  the  deliciency. 


124                      SINGLE  RULE  OF  THREE  DIRECT.         Sect.  II.  8. 

22.  If  I  buy  76  yards  of  cloth  for  23.    A    man    spends    g3,25    per 

till  13,17,  what  did  it  coat  per  Ell  week,  what  is  that  p^r  annum  ? 

English?                     Aks.  ^lyUQU  Am.  ^iG9,-le>4. 


21.  if  3  liorses  :ind  4  oxen  be  worth  9  cows,  how  many  cows  will  8 
horses  and  8  oxen  be  worth  :  /Ins.   I H. 


25.  Bougl^t  a  silver  cljp,  weighing  9oz.  Apn-t.  Xf^^rt.  for  £3  2^.  Zd.  o|.j. 
what  was  that  ner  otmca  l  Arts.  '6i.   'dd. 


Sec*.  II.  8.        SINGLE  RULE  OF  THREE  DIHECT. 


126 


26.  There  is  a  cistern  which  has 
4  cocks,  the  first  will  empty  in  10 
minutea,  the  second  in  20  minutes, 
the  third  in  40  minutes,  and  the 
fourth  in  80  minutes  ;  in  what  time 
will  all  four  running  together  empty 
it? 


Min. 

Cist. 

( 10     Cist.         Min. 

(6 

)20  :   1     :   :     60  : 

)3 

yio 

)],5 

(so 

(    ,75 

In  1  hour  the  4  cocks 

would  empty   -     -     -     - 

11,25  Cw 

Then, 

Cist.     Min.     Cist. 

Min. 

As  11,25  :  60  :  :   1   • 

5,33  Ans, 

27.  A  man  having  a  piece  of  land 
te  plant,  hired  two  men  and  a  boy  to 
plant  it,  one  of  the  men  couM  plant 
it  in  12  days,  the  other  in  15  days, 
and  the  boy  in  27  days  ;  in  how  long 
time  would  they  plant  it  if  they  all 
worked  together  ? 

Ans.  5t2'l6  days. 


i 


28.  A  merchant  bought  270  quin- 
tals of  cod  lish,  for  ;f5780 ;  freight 
§37,70  ;  duties  and  other  ^charges 
^30,60  ;  what  must  he  sell  it  at  per 
quintal  to  gain  ^143  in  the  whole  ? 
Ans.  $3,671. 

The  sum  of  all  the  expenses  of 
the  tish  with  the  Merchant's  gain 
must  be  found  for  the  secocd  term. 


29.  If  a  staff  5/r.  8m.  in  length 
cast  a  shadow  of  6  feet ;  how  higli  is 
that  steeple  whose  shadow  measures 
153  feet?  Ans.  U4yeet. 


ii-^ 


H 


^      tic 


116  oiNGLE  RULE  OF  THREE  DIRECT.        Sect.  II.  8. 

30.  Bought  12  pieces  of  cloth  each        31.  Bought  4  pieces  of  Holland, 

10  yards,  at  g  1,73  per  yard,  what  each  containing  24  Ells  English,  for 

came  they  to  ?  g96  ;  how  much  was  that  per  yard  ? 
Ans.  ;J210.  Ans.  QO  cents. 


32.  Bought  9  chests  of  tea,  each  weighing  3C.  2grs.  Zllb.  at  £4  9#. 
per  cwt.  what  came  they  to  ?  Am.  J£147  13*.  B^d. 


Sect.  II.  8.        SINGLE  RULE  OF  THREE  DIRECT.  127 

33.  A  bankrupt  owes  in  all  972  dollars,  and  his  money  and  effects  are 
W  ^607,50 ;  what  will  a  creditor  receive  on  $11 ,333  ?        Ms.  $7,083 


I 


34.  Bought  126  gallons  of  mm  for  $110,  how  much  water  must  be  added 
to  it  to  reduce  the  first  cost  to  75  cents  per  gallon?^  Ans.  20|ga/. 


36.  A  owes  B  £3475,  but  B  com-  36.  If  a  person  whose  rent  is 
pounds  with  him  for  13s.  4d.  on  the  $145  pays  $12,63  of  parish  taxes, 
pound  ;  what  must  he  receive  for  how  much  should  a  person  par 
his  debt  ?        Am.  £2316  13j.  4d.        whose  rent  is  $378  ? 

Ans.  $32,925. 


I'ia  SINGLE  iiULE  OF  THKEE  iN VERSE.        Sect.  H.  8. 


[nvcrsc  Froportion. 

IN  some  questions  the  number  sought  becomes  less,  when  the  circum- 
stimccs  from  which  it  is  do ri veil  become  greater.  Thus  wlien  the  price  ot' 
{roods  iiicrciise  the  qnantily  which  may  be  bought  ibr  a  given  sum,  is  smaller 
When  the  number  of  men  employed  at  work  is  increased,  the  time  in  which 
Ihey  may  complete  it  becomes  shorter  ;  and  when  the  activity  of  «ny 
cauie  is  increased,  the  quantity  necessary  to  produce  any  given  effect  is 
di.^lini!:•hed. 

These  and  the  lilce  cascj  belong  to  the 

SINGLE  RULE  OF  THREE  INVERSE. 

The  Single  Rula  of  Three  Inverse  teaches  by  having  three  numbers 
eivcn  to  find  a  fourth,  having-  the  same  proportion  to  the  second,  as  the 
fir:«t  has  to  the  third. 

If  more  require  less,  or  less  require  more,  the  question  belongs  to  the 
Single  Rule  of  Three  Inverse. 

Mirrc  rujuirin^  Icsi,  is  when  the  third  term  is  greater  than  the  first,  and 
requires  the  fourth  term  to  be  less  than  the'second. 

IjC^s  rcijuiri/ig  more,  is  when  the  third  term  is  IccS  than  the  first,  and  re- 
quires the  fourih  term  to  be  (greater  than  the  second. 

RULE. 

"  State  and  reduce  the  terms  as  in  the  rule  of  three  direct ;  then  multi- 
ply tlic  iir-st  and  second  ti.rms  together,  divide  the  product  by  the  tliird, 
and  the  quoiient  will  be  the  ansv.er  in  the  same  denomination  with  tlic 
&ccfuid  term." 

EXAMPLES.    " 

1.  If -18  men  build  a  wall  in  iii  days,  hov/  many  men  can  do  the  same  in 
1P2  days  t 

OPKUATIO.V. 

Men.     Days>     Aleiu  Here  the  third  term  is  greater  than  me 

Aa  48    :    24  :  :    192  first,  and  coaimon  sense  teaches  the  fourth 

43  term,  or  answer  m<ibt  be  las  than  the  sec- 

— —  pnd  ;  for  if  4l>  men  can  do  the  work  in  2  1 

102  4ay3,  certainly  192  men  will  do  it  in  less 

9G  time.     In  this  way  it  may  be  determined  ^ 

— -  if  .a  queslioii  Jaelooii-  to  the  Rule  X)f  Three  i 

Id2)\l52^r>  answer.  Inverse.                                                            i 

£.  !f  a  board  be  9  inches  broad,         3.  How  many  yards  of  sarcenet,  | 
how  much   in  length  will   make  a     "qrs.  wide,  will  line  9  yards  of  cloth, 
square  fool  i        '7i}i,s.  IC  inches.  ofS^r*.  wide?  Ans,  24  yards.    _, 


Sect.  II.  8.        SINGLE  RULE  OF  THREE  INVERSE. 


129 


4.  Lent  a  friend  292  dollars  for 
6  months  ;  some  time  afterwards  he 
lent  me  806  dollars  :  how  long  may 
I  keep  it  to  balance  the  favor  ? 

Jins.  2  months^  5  days. 


5.  A  garrison  had  protision  for  8 
months,  at  the  rate  of  15  ounces  to 
each  person  per  day  ;  how  much 
must  be  allowed  per  day  in  order  that 
the  provision  may  last  9^  months  ?  ^ 
Jlns.  12|2.  ounces. 


6.  A  garrison  of  1200  has  pro- 
visions for  9  months  at  the  rate  of 
14  ounces  per  day,  how  long  will 
the  provisions  last  at  the  same  al- 
lowance if  the  garrison  be  reinforced 
by  400  men  ?         Jinx.  6  *  months. 


7. Flow   must   the   daily 

allowance  be  in  order  that  the  pro- 
visions may  last  9  months  after  the 
garrison  is  reinforced  ? 

Jlns.  10^  ouncet. 


6.  How  much  land  at  ^2,50  per 
acre  should  be  given  in  exchange  for 
360  acres  at  $3,75  per  acre  .' 

Jlns.  540  urrc. 


9.  What  sum  should  be  put  to  in- 
terest to  gain  as  much  in  1  month  aa 
^127  M'ould  gain  in  12  months  ? 

Ans.  $1521. 


R 


150 


SINGLE  RULE  OF  THREE  INVERSE.       Sect.  II.  8. 


10.  If  a  man  perform  a  journey  in 
13  days,  when  the  day  is  li  hours 
loiijr,  in  how  many  will  he  do  it 
when  the  day  is  but  10  hours  ? 

dns.  1 3  days. 


11.  If  a  piece  of  land  40  rods  in 
length,  and  4  in  breadth  make  an 
acre,  how  wide  must  it  be  when  it  is 
but  25  rods  long  ? 

Ans.  G^rods. 


["J.  There  wns  a  certain  building 
tr.ir^i?d  in  G  months  by  120  workmen, 
but  the  sjunc  being  demolished,  it  is 
it'tiuircd  to  be  built  in  two  months  ; 
i  demuid  how  many  men  must  be 
(TU'loved  about  it  ?      Ans.  480  men 


13.  How  much  in  length,  that  ia 
3  inches  broad,  will  make  a  square 
foot  ?  Jns.  48  inches. 


14.  There  is  a  cistern  having  1 
pipe  which  will  empty  it  in  10  hours, 
how  many  pipes  of  the  same  capacity 
will  empty  it  in  24  minutes  ? 

Ans.  26  pipes. 


15,  If  a  field  will  feed  6  cows 
91  da3's,  how  long  will  it  feed  21 
cows  1  Ans.  26  days. 


16.  If  thequiirtern  loaf  weigh  4i 
pounds  when  wiieat  is  ^2  per  bush- 
el, ;*liat  nin?t  it  weigh  when  wheat 
!>  ^],riO  the  bushel? 

Ans.  Gib. 


17.  How  many  yards  of  baize,  3 
quarters  wide,  will  line  a  cloak 
Avhich  has  in  it  12  yards  of  camblet, 
half  yard  wide  ? 

Ans.  8  yards. 


Sect.  II.  8         SINGLE  RULE  OF  THREE  INA'ERSE.  131 

GENERAL  RULE 

For  stating  all  qnestimis  n^hethcr  direct  or  inverse. 

1.  Place  that  number  for  the  third  term,  nhich  signifies  the  same  kind 
•of  thing,  with  what  is  sought,  and  consider  whether  the  number  sought  will 
be  greater  or  less.  If  greater,  place  the  least  of  the  other  terms  for  the 
first ;  but  if  less,  place  the  greater  for  the  first,  and  the  remaining  one  for 
the  second  term. 

Multiply  the  second  and  third  terms  together,  divide  th^  product  by  the 
first,  and  the  quotient  will  be  the  answer. 

EXAMPLES. 

1 .  If  30  horses  plough  12  acres,  how  many  will  10  plough  in  the  same  time? 

OPERATIOIVS. 

//.     H.      Aq.  Here  because  the  thing  sought  is  a  number  of 

30  :  40  :  :  12  acres,  we  place  12,  the  given  number  of  acres, 

12  for  the  third  term  ;  and  because  40  horses  will 

■  plough  more  than  12,  we  make  the  lesser  num- 

30)480(10  Ans.       ber,  30,  the  first  term,  and  the  greater  number, 
40,  the  second  term. 

2.  If  40  horses  be  maintained  for  a  certain  sum  on  hay  at  5  cents  per 
stone,  how  many  will  be  maintained,  on  the  same  sum,  when  the  price  of 
hay  rises  to  8  cents  per  stone  ? 

C.     C.      H.  Here,  because  a  number  of  horses  is  sought, 

8   :  5  :  :  40  we  make  the  given  number  of  horses,  40,  the 

40  third  term,  and  because  fewer  will  be  maintaia- 

ed  for  the  same  money,  when  the  price  of  hay 

8)200(25  Ans.         is  dearer,  we  make  the  greater  price  8  cent>i, 

16  the  first  term,  and  the  lesser  price,  5  rent5,  the 

—  second. 
40 
40 

The  first  of  these  examples  is  Direct,  the  second  Inverse. 

Every  question  consists  of  a  supposition  and  a  demand. 

In  the  first  the  supposition  is,  that  30  horses  plough  12  acres,  and  the  de- 
mand horv  many  40  nnll  plough  ?  and  the  first  term  of  the  proposition,  30. 
is  found  in  the  supposition  in  this  and  every  ofher  direct  question. 

In  the  second,  the  supposition  is  that  40  horses  are  maintained  on  hay  at 
5  ceiUs  per  stone,  and  the  demand,  how  many  az//  be  maintai^ied  on  hay  at  8 
cents?  and  the  first  term  of  the  proportion,  8,  is  found  in  the  demanJ,  in 
this  and  every  other  inverse  qtiestiou. 

3.  If  a  quarter  of  v.'hcat  aObrd  GO  4.  If  in  12  months,  100  doUar^i 
tenpenny  loaves,  how  many  eight-  gain  6  dollars  interest,  what  will  gain 
penny  loaves  may  bo  obtained  from     the  same  sum  in  5  months? 

it  ?  Ans.  75  loaves.  Ans.  240  dollars. 


132  SUPPLEMENT  TO  THE  SING.  R.  OF  THREE.     Sect.  H.  8. 


SUPPLEMENT  TO  THE  SINGLE  RULE  OF 

THREE. 


qUESTlONS. 
1.  What  is  the  Single  Rule  of  Three  ;  or  the  Rule  of  Proportion  ? 
'J.  How  many  Ivinds  of  Proportion  are  there  ? 

3.  What  is  it  that  the  Single  Rule  of  Three  Direct  teaches  ? 

4.  How  can  it  be  known  that  a  question  belongs  to  the  Single  Rule  of 

Three  Direct  ? 

5.  What  is  understood  by  more  reriuiring  more  and  less  requiririg  less  ? 
(i.  How  are  questions  in  the  Rule  of  Three  stated  ? 

7.  Having  stated  the  question,  how  is  the  answer  found  in  direct  pro- 

portion ? 

8.  What  do  you  observe  of  the  fjrst  and  third  terms  concerning  the 
tiiflcrent  denominations,  sometimes  contained  in  them  ? 

L^..  Whc-n  the  second  term  contains  different  denominations,  what  is  to 
be  done  ? 

10.  How  is  it  known  what  denomination  the  quotient  is  of? 

11.  If  the  quotient  or  answer  be  found  in  an  inferior  denomination,  what 

io  to  bo  done  ? 

12.  W4^en\t]ie  terms  are  given  in  Federal  Money,  how  is  the  operation 

conducted  ? 

13.  How  arc  the  sums  in  Federal  Money  reduced  to  the  same  denomi- 

nation ? 

14.  V»'hen  .sny  number  of  barrels,  bales,  pieces,  &c.  are  given,  what  is 

the  method  of  procedure  ? 
\h.  What  is  it  that  the  Single  Rule  of  Three  Inverse  teaches  ? 
lb'.  How  are  the  questions  stated  in  Inverse  Proportion  ? 
1?.   What  is  understood  by  more  requiring  less  and  less  requiring  more  ? 
IH,  How  is  the  answer  found  in  the  Rule  of  Three  Inverse  ? 
1'?.  What  is  the  general  Rule  for  stating  .ill  questions,  whether  Director 

Inverfce  ? 

EXERCISES. 

■   l!  liiy  horsff  and  saddle  are  worth  13  guine.is,  and  my  horse  be  worth 
•:  .  'lie-  >o  ui;\i;it  ;.s;  the  saddjc',.  prav  what  i.s  the  value  of  mv  horse  ? 

Ant.  T'j'dollurs. 


Sect.  II.  8.     SUPPLEMENT  TO  THE  SING.  R.  OF  THREE.  133 

2.  Howmany  yards  of  matting  that        3.    Suppose    800    soldiers  were 

IS  half  a  yard  wide  will  cover  a  room     placed  m  a  garrison,  and  their  pro- 

that  is  18  feet  wide  and  30  feet  long  ?     visions  were  computed  sufficient  for 

Ans.  ]20  yards.         two    months  ;    how   many   soldiers 

must  depart  that  the  provision  mny 

serve  them  6  months  ?    Ans.  480. 


4.  I  borrowed  185  quarters  of  corn  when  the  price  was  19s.  how  much 
must  I  pay  to  iademnily  the  lender  when  the  price  is  17s.  4d.  1 

Ans.  2021^. 

6.  Bought45barrelsof  beef  at  ^3,50 
per  barrel,  among  which  are  1 6  barrels, 
whereof  4  are  worth  no  more  than  3  of 
the  others  ;  how  much  must  I  pay  ? 
_  Ans.  ^143,60. 


1 


134  SUPPLEMENT  TO  THE  SING.  R.  OF  THREE.     Sect.  U.  8. 

6.  A  and  B  depart  for  the  same  place  and  travel  the  same  road  ;  but  A 
goes  5  days  before  B  at  the  rate  of  20  miles  per  day  ;  B  follows  at  the 
rate  of  23  miles  per  day  ;  in  what  time  and  distance  will  he  overtake  A  ? 
•fln*.  B  will  overtake  A  in  20  days,  and  travel  500  miles. 

Here  two  st;«tements 
will  be  necessary,  one 
to  ascertain  the  time, 
and  the  other  to  ascer- 
tain the  distance. 


Method  of  assessing  town  or  parish  taxes. 

'  1.  An  inventory  of  the  value  of  all  the  estates,  both  real  and  pcr^nal, 
and  the  number  of  polls  for  which  each  person  is  rateable,  must  be^^^Ji-en 
in  separate  columns.  Then  to  know  what  must  be  paid  on  the  dollar,  make 
the  total  value  of  the  inventory  the  first  term  ;  the  tax  to  be  assessed  the 
second  ;  and  1  dollar  the  third,  and  the  quotient  will  shew  the  value^n 
the  dollar. 

NoTit.    Tliis  method  is  taken  from  Mr.  Pike's  Arithmetic,  with  this  difference,  that  here 
the  money  is  reduceU  to  Federal  Curreucy. 


Sect.  II.  8.     SUPPLEMENT  TO  THE  SING.  R.  OF  THREE. 


135 


2.  Malce  a  table,  by  multiplying  the  value  on  the  dollar  by  1,  2,  3,  4,  5,  &c. 

3.  From  the  Inventory  take  the  real  and  personal  estates  of  eac^j  man, 
and  find  them  separately,  in  the  table,  which  will  shew  you  each  man's 
proportional  share  of  the  tax  for  real  and  personal  estates. 

If  any  part  of  the  tax  be  averaged  on  the  polls,  before  stating  to  find  the 
value  on  the  dollar,  deduct  the  sum  of  the  average  tax  from  the  whole  sum 
to  be  assessed  ;  for  which  average  make  a  separate  column  as  well  as  for 
the  real  and  personal  estates. 

EXAMPLES. 

Suppose  the  Geaoral  Court  should  grant  a  tax  of  130000  dollars,  of  which 
a  certain  town  is  to  pay  ^3250,72  and  of  which  the  polls  being  624  are  to 
pay  75  cents  each  ;  the  town's  inventory  is  G9568  dollai-s  ;  what  vnW  it  be 
on  the  dollar  ;  and  what  is  A's  tax  (as  by  the  inventory)  whose  estate  is  as 
follows,  viz.  real,  856  dollars  ;  personal,  103  dollars  ;  and  he  has  4  polls  ? 
Pol.     Cls.      Pol.     Dolls. 

1.  As,  1     :    ,75  :  :  624  :  468  the  average  part  of  the  tax  to  be  de- 
diicted  from  ^3250,72  and  there  will  remain  J^2782,72. 

Dolh.     Dolls.  Cts.     Dolls.  Cts. 

2.  As  G9568   :  2782,72  :  :   1     :     4  on  the  dollar. 

TABLE. 


Dolls. 

Dolls,  cts. 

Dolls. 

Dolls 

.  cts. 

Dolls.     Dolls 

1  is 

4 

20  is 

80 

200  is       8 

2  — 

8 

30  — 

1 

20 

300—    12 

3  — 

12 

40  — 

1 

60 

400—    16 

4  — 

16 

50  — 

2 

00 

500  —    20 

5  — 

20 

60  — 

o 

40 

600  —   24 

6  — 

24 

70  — 

2 

80 

700  —   28 

7  — 

28 

80  — 

3 

20 

800  —   32 

8  — 

32 

90  — 

3 

60 

'      GOO  —   36 

9  — 

36 

100  — 

4 

00 

1000—    40 

10  •— 

40 

Now  to  find  what  A's  rate  will  be. 

His  real  estate  being  856  dollars,  I  find  by  the  Ta 
ble  that  800  dollars  is  g532  cts. 
that  50  —      —  2 

that     6  —      —  0  24 


Therefore  the  tax  for  his  real  estate  is  34  24 
In  the  like  manner  I  find  the  tax 

for  his  personal  estate  to  be 

His  4  polls,  at  75  cents  each,  are       3 


"I 


4    12 


pA  36 


I     Real. 
\Dolls.  Cts. 


34 


Personal 
Dolls.  Cts 


12 


Polls. 
Dolls.  Cts. 


Total. 
Doils.  Cts. 


41 


36 


1 30  DOUBLE  RULE  OF  THREE.  Sect.  II.  9. 

^  0.  DOUBLE  ]U  LE  OF  THREE. 


THE  Double  Rule  of  Throe,  sometimes  called  Compound  Proportion, 
li'aclies  by  havint;  five  nnmhei-s  i;ivpn  to  find  a  sixth,  which,  if  the  pro- 
))ortion  be  direci,  Uiust  bear  the  same  pioportion  to  tlie  fourth  and  fifth  as 
the  thiid  does  to  the  fi^^t  and  second.  But  if  tlie  proportion  be  inverse, 
tlie  sixth  number  must  bear  the  same  proportion  to  the  fourth  aud  fifth,  as 
the  first  does  to  the  second  and  third. 

RULE. 

1.  "  Slate  the  question,  bv  placing  the  three  conditional  terms  in  such  or- 
der (ha(  that  number  whicli  is  the  cause  of  gain,  loss,  or  action,  may  possess 
tlie  first  place  ;  that  which  denotes  space  of  time,  or  distance  of  place,  the 
second  ;  and  that  which  is  the  .1,'ain,  Joss,  or  action,  the  tliird." 

2.  •'  Place  the  other  two  terms,  which  move  the  question,  under  those 
of  the  same  name." 

3.  "  Then,  if  the  blank  place,  or  term  sought,  fall  under  the  (bird  jtlare, 
the  propoiiion  is  direct,  tlierefore,  multiply  the  thieo  last  terms  together, 
for  a  dividend,  and  the  other  two  for  a  di\isor  ;  then  the  quotient  will  be 
the  answer." 

4.  *'  But  if  the  blank  f:dl  under  the  first  or  second  place,  the  prcj»orlion 
is  invcr-so,  wherefore  muUij)ly  the  first,  second  and  last  terms  t0G;cther,  for  a 
dividend,  and  the  other  two,  for  adiVisor  ;  the  "quotient  will  be  the  answer." 

EXAMPLES. 

If  100  dollars  gain  6  dollars  in  12  nionths,  what  will  400  dollars  gain  in 
C  months  ? 

Statement  of  the  uuestion, 
D.        M.        D. 

KJO     :     12    :  :   0  Terms  in  the  supposition,  or  conditional  tenns. 
400     :       8  Terms  Xi'hich  move  the  nuestion. 

Of  the  three  conditional  terms,  it  is  evident  that  100  dollars  put  at  inte- 
rest, is  that  one  uhich  is  the  cause  of  gain  ;  consequently  100  dollars  must 
be  the  first  term  ;  and  because  12  months  is  the  space  of  time  in  which  the 
jiain  is  made,  this  must  be  tl)e  second  term  ;  and  G  dollars  which  is  the  gain, 
liie  third  term.  The  other  two  ierziis  must  then  be  arranged  under  those 
of  tlie  same  name. 

Now  as  the  blank  fills  under  the  third  place,  therefore,  the  question  is 
in  direct  proportion,  and  the  answer  is  found  by  multiplying  the  three  last 
ternw  together  for  a  dividend,  and  the  two  first  for  a  divisor. 


Then,  12|00)192l00( 
Dolls.  leArns. 


1 200  Divisor.  1  f -£  00  Dividend. 

2.  If  JOO  dollars  gain  C  dolljirs  in  12  montLs,  \u  wh;,t  time  will  4P0  dol 
lars  gain  16'!' 


OPERATION 

100  :  12  :  : 
400   8 
8 

100 
12 

3200 
6 

Sect.  II.  0.  DOUBLE  RULE  OF  THREE.  137 

OPERATION. 

D.    M.      D. 

100  :  12  :  :  6         Here  the  blank  falling  under  the  second  term,  the 
400  16     proportion  is  indirect. 

6  12         Therefore  multiply  the  first,  second  and  last  terms 

together  for  a  dividend,  and  the  other  two  for  a  divisor. 


2400  div.    192 
100 


10200  dividend.  M. 

Then  24|00)192|00(8  Answer. 
192 

3.  A  Farmer  sells  £04  dolls,  worth         4.  If  7  men  can  reap  84  acres  of 
of  grain  in  5  years,  when  it  is  sold     wheat  in   12  days,  how  many  men 
;it  GO  cents  per  bushel,  ^vhat  is  it     can  reap  100  acres  in  3  days  ? 
jiL-r  bushel  wlien  lie  sells  1000  dolls, 
worth  in   18  years,  if  he  sell  the 
sawe  quantity  yearly  ? 

Os.     Y.       D. 
CO  :   5  :  :  204       cts.  m. 
IB  :  :  1000  :  ,816  Am. 


M. 

D. 

A. 

7  : 

12  : 

:84 

M. 

6: 

:  100 

•  20  Alls 

5.  If  a  family  of  9  persons  spend  450  dollars  in  5  months,  Ifow  much 
^vould  be  sufficient  to  maintain  them  8  months,  if  5  persons  more  were 
added  to  the  family  ?  .  Ans.  $1120. 


138  SUPPLEMENT  TO  THE  DOUB.  R.  OF  THREE.     Sect.  II.  9. 

SUPPLEMENT  TO  THE  DOUBLE  RULE  OF 

THREE. 


QUESTIONS. 

1.  What  is  the  Double  Rule  of  Three  ;  or  Compound  Proportion? 

2.  How  are  quc?fions  to  be  stated  in  the  Double  Rule  of  Three  ? 

3.  How  is  it  known  after  the  statement  of  the  question,  whether  the 

Proportion  be  Direct  or  Inverse  ? 

4.  When  the  Proportion  is  Direct,  how  is  the  answer  to  be  found  ? 
6.  When  the  Proportion  is  Inverse,  how  is  the  answer  to  be  found  ? 

EXERCISES. 

1.  If  6  men  build  a  wall  20  feet  long,  6  feet  high  and  4  feet  wide  in  16 
days,  in  what  time  will  24  men  build  one  200  feet  long,  8  feet  high  and  G 
thick?     Ms.  80  days. 

The  solid  contents 
in  each  piece  of  wall 
according  to  the  given 
dimensions,  must  be 
found  before  stating 
the  question. 

i.'.  If  40/w.  at  Boston  make  3G  at 
Amsterdam,  and  90/A.  at  Amsterdam 
make  116  at  Dantzick,  how  many 
lb.  at  Boston  are  equal  to  260/6.  at 
Dantzick  ?  Ans.  224^/6. 

N.  B.  The  answer  to  this 
question  is  found  by  two 
statements  in  the  Rule  of 
Three  Direct. 


Sect.  II.  9.     SUPPLEMENT  TO  THE  DOUB.  R.  OF  THREE.  139 

3.  If  the  freight  of  UCtvt.  2qrx.  6/6.  275.  miles  cost  ^27,78  ;  how  far 
mny  eOCzi't.  ^xjr.i  be  bhipped  for  C234,78  ?  Jns.  480  miles. 


4.  An  usurer  put  out  75  dollars,  6.  If  7  men  can  make  81  rods  of 

at  interest,   and  at  the  end  of  eight  wall  in  6  days  :  in  what  time  will  10 

months   received  for  principal   and  men  make  160  rods  ? 

interest,    79  dollars  ;    I  demand  at  Ans.  7i  days. 
what  rale  per  cent  he  received  in- 
terest ?                      Ans.  8  per  cent. 


140  SUPPLEMENT  TO  THE  DOUB.  R.  OF  THREE.    Sect.  II.  9. 

6.  If  the  freight  of  9hhds.  of  sujar,  earh  wei^^hio*^  I9rw  or,  u„ 


mg  .1  C«»^  100  leagues  ?  ,4„,^  ^^^2  1 1,. 


10%d'. 


Sect.  II.   10. 


PRACTICE. 

^  10.  PRACTICE. 


141 


"  Practice  is  a  contraction  of  the  Rule  of  Three  Direct,  when  the  first 
term  happens  to  be  an  unit  or  one  ;  it  has  its  name  from  its  daily  use  among' 
Merchants  and  Tradesmen,  being  an  easy  and  concise  method  of  working 
finest  questions  which  qccur  in  trade  and  business." 

Proof.  By  the  Single  Rule  of  Three,  Compound  Multiplication,  or  by 
varying  the  parts. 

Before  any  advances  are  made  in  this  rule,  the  learner  must  commit  to 
memory  the  following 

TABLES. 


ALIQUOT,    OR    EVEN    PARTS 

Parts  of  a  shill.'    of  a  £. 


d. 

s. 

6 

is 

1 

4 
3 
2 

— 

J. 

3 
k 
i 

H 
1 

— 

i 

— 

3. 

, 

_L 

and       jj. 


4  8  0 
_Jl_ 


5d.  is  the  sum  o{4d.  4'  id. 
■  7c/.         G(/.  4'  Id. 

8rf.      is  twice      id. 

9d.  is  the  sum  of  6(/.  ^  3d. 

\0d.        Gd.^-Ad. 

lid.        Gd.3d.4'2d. 


Pts. 

s. 
10 

6 

5 

4 

3 

2 

1 

1 

1 

1 

0 

0 

0 

0 


OF  MONEY. 

of  a  pound.      Practice  admits  of  a  great  va- 

d.  is    £,'  riety  of  cases,  the  multiplicity 

0  —     ^  of  which  serves  little  else  than 

8  —      i  that  of  confounding  the   mind  of 

0  —      A  the  ffcholar  ;  a  different  method 

0  —      i  will  be  pursued  here,  and  the 

4  —      J.  whole  comprised,  in  a  few  cases, 
G  i  such  as  shall  be  useful  and  easy 

5  —    _!_  for  the  scholar  to  bear  in   his 


4 

3 

0 

10 


oi 


_!_  memory, 

jL  The  small  number  of  exam- 
j_  pies  under  each  case  will  be 
_i_  made  up  in  the  Supplement ; 
_*_  this  will  lead  the  scholar  to  a 
_i_  more  particular  consideration  of 
1   them. 

9S 


OPERATIONS. 


POUNDS,  SHILLINGS,  PENCE,  FARTII.  DOLLARS,    CENTS,    MILLS. 

When  the  price  of  the  given  quan-  RULE, 

tity  is  j£l.   Is.   Id.  per  pound,  yard.        Multiply  the  quantity  by  the  price 
&c.  then  will  the   quantity  itself  be    of  one  pound,  yard,  &:c.  the  product 
the  answer  at  the  supposed  price. —   will  be  the  answer. 
Therefore, 

CASE  1. 

When  the  ■price  of  1  ynrd,  pounds  ^c. 
consists  of  faj-things  onh; ;  If  it  be  one 
farthing,  take  a  fourth  part  of  the 
quantity  ;  if  a  half  penn}',  take  a  half ; 
if  three  firthings,  take  a  half  and  a 
fourth  of  tiie  quantity  and  add  them. 
Thi-?  gives  the  value  in  pomre.  which 
mu?t  bw  it'ducod  to  pounds. 


142 


PRACTICE. 


Sect.  II.  10. 


POUNDS,  SHILLINGS,  PENCE,  FARTH. 

EXAMPLES. 
1.    What  will  362  yards  cost  at 
iJ.  per  yard  ? 

OPERATION. 

2)362 
n)\Q]  pence. 


155.  Id.  Jlns. 

Here  the  quantity  stands  for  the 
price  at  one  penny  per  yard,  but  as 
two  farthings  are  but  half  one  penny, 
therefore  dividing  the  quantity  by  2, 
gives  the  price  at  half  a  penny  per 
yard,  which  must  be  reduced  to  shil- 
lings. 

2.  What  will  354A  yards  cost  at 
■^d'  per  yard  ?  ^ 

OPERATIOX.    H 

d.         q. 

4)354       2 


12)80  2 

7s.  Ad,  2  .ins. 

3.    What  will  263  yards  cost  at 

Zq.  per  yard  ?  Ans.  1 6s.  5^^. 


DOLLARS,    CENTS,    MILLS. 

1.  What  will  362  yards  cost  at 
7  nulls  per  yard  ? 

OPERATION. 

3  G   2  quantity. 
,0  0  7  price. 

^2,  6  3  4  Ans. 

Note.  The  answers  in  the  different 
kinds  of  money  will  not  always  com- 
pare, because  in  the  redaction  of  the 
price,  a  small  fraction  is  often  lost  or 
sained. 


2.  What  will  3541  yards  cost  at  3 
mills  per  yard  ? 

operation; 
3  5  4  ,5  quantity. 
,0  0    3  price. 


gl   ,0  6  3    5 

3.  What  will  263  yards  cost   at 
1  cent  per  yard  ?         Ans.  ^2,63. 


4.  What  will  816  yards  cost  at  \q. 
per  yard?  Ans.  17s. 


4.    What  will  816  yards  cost  at 
3  mills  per  yard  ?         Ans.  $2,448. 


Sect.  II.  10. 


PRACTICE. 


143 


POUNDS,  SHILLIN-GS,  PENCE,  FARTH. 

5.  What  will  97  yards  cost  at  3q. 
per  yard  ?  Ans.  6s.  O^d. 


6.  What  will  126  yards  cost  at  id. 
per  yard  ?  Jins.  5s.  3d. 


CASE  2. 

When  the  price  of  lib.  lyd.  ^-c. 
consists  of  pence,  or  of  pence  and 
farthings;  if  it  be  an  even  part  of  a 
shilling,  find  the  value  of  the  given 
quantity  at  Is.  per  yard,  (the  quan- 
tity itself  expresses  the  price  at  Is. 
per  yard  ;  if  there  are  quarters,  iait 
write  for  i  3d.  for  i  6d.  for  |  9d.) 
and  divide  by  that  even  part  which 
the  price  is  of  1  shilling.  If  the 
price  be  not  an  aliquot  or  even  part 
of  one  shilling,  it  raust  be  divided 
into  two  or  more  aliquot  parts  ;  cal- 
culate for  these  separately,  and  add 
the  values  ;  the  answer  will  be  ob- 
tained in  shillings,  which  must  be 
reducct!  to  pounds. 


DOLLARS,    CENTS,    MILLS. 

5.  What  will  97  yards  cost  at  1 
cent  per  yard  ?  Ans.  ,97c<*. 


6.  What  will  126  yards  cost  at  7 
mills  per  yard  ?  Am.  $0,882. 


144 


PRACTICE. 


Sect.  IT.  10 


rov.VDS, 

SHILLINGS,  PENCE,  FARTH. 

DOLLARS,    CENTS,    WILLS. 

EXAMPLES. 

7.  What 

will  ■476  3'aril9  come  to  at 

1 .  What  will  476  yards  cost  at  7^. 

10  cents  4 

mills  per  yard  ? 

per  yard  ? 

OPERATION. 

OPERATION. 

476 

S. 

,104 

Cd.       1 

476  Price  at  Is.  per  yard. 

llrf.  1 

238  Price  at  Gd.  per  yard. 

1904 

59  Gd.  price  I  !^d.  per  yard. 

4760 

2|0)29(7  Gd.  price  at  7!  per  yd. 

^49,504  Ans.  , 

i 

l]4   lis.   Gd.  Ms. 

FROOF. 

PROOF. 

1.  By  the  Rule  of  Three. 
F.        £.        s.        d.        Y. 

A9  47G  :  14    17    G  :  :  1 
20 


297 
12 


47G)3570(7(/. 
3332 

238 
4 


)952(2^r. 
952 


2.  By  Compound  Multipli- 
cation. 
£.     s.'   d. 

7^  price  of  1  yard. 
10 


6     3  price  of  10  yards. 
10 


3     2     6  price  of  100  yards. 
4 


12  10  0  price  of  400  yards. 

2     3  9  price  of  70  yards. 

3  9  price  of  G  yards. 

JL'l'f    17  C  price  (f  4." 6  yards. 


cts.  m.  D.  cts.  m.  yds. 
,10  4)4  9  50  4(476 
4  1   6 


7  9  0 

7  2  8 

6  2  4 

6  2  4 


Sect.  H.   10. 


PRACTICE. 


14& 


POUNDS,  SHILLINGS,  i^ENCE,  FARTH. 

What  will  176  yards  cost  at  9iJ. 
per  yard  ? 

OPERATION. 

s. 
[  6ii.     i  I  176  value  at  Is.  per  yard. 

I  3.'^     I  j  88  value  at  6d.  per  yard. 

i  ii.     I  of  44  value  at  3rf.  per  yard. 

7  4d.  val.  at  -kd.  per  yd. 

2|0)13!9  4c/.— atgiJ.  per  yd. 
£6   I9s.  4d.  Ans. 

FllOOF. 


I 


i 


3.    What  will  568^  yards  cost  at 
7<^.  per  yard  I      Ans.  £,16  lis.  S^d. 


DOLLABS,    CENTS,    MILLS. 

8.  What  will  176  yards  cost  at  13 
cents,  2  mills,  per  yard  ? 

Ans.  g23,232. 


9.  What  will  568^  yards  cost  at 
9  cents  7  mills  per  yard  ? 

Ans.  |55,12. 


146 


PRACTICE. 


Skct.  II.  10. 


POUNDS,  SHILLINGS,  PENCE,  FARTH. 

4.  What  will  6862  yards  come  to 
at  2|d.  per  yard  ? 

Jlns.  £7  2s.   101(7. 


DOLLARS,    CENTS,    MILLS. 

10.  What  will  685^-  yards  come 
to  at  3  cents  5  mills  per  yard  ? 

^«j,  $24,001. 


5.  What  will  649}  yards  cost  at 
iOc/.  per  yard  ?     Ans.  £21  1$.  O^d. 


6.  What  will  6833  yards  cost  at 
8ic/.  per  yard  ?    Ans.  £23  10s.  03^. 


11.  What  will  6491  yards  cost  at 
13  cents  9  mills  per  yard  1 

Ans.  $90,245. 


12.  What  will  683f  yards  cost  at 
1 1  cents  7  mills  per  yard  ? 

Ms.  79,998 


Sect.  II.  10. 


PRACTICE 


147 


POUNDS,  SHILLINGS,  FENCE,  FARTH. 
CASE    3. 

If  the  price  of  lib.  lyd.  ^c.  be 
sJiillings  and  pence  and  an  even  part 
of  £,1,  Divide  the  value  of  the  given 
quantity  at  £l  per  yard  by  that  even 
part,  which  the  price  is  of  i;l.  The 
quotient  will  be  the  answer. 
EXAMPLES. 

1.  What  will  7191  yards  cost  at 
Is.  4c?.  per  yard  ? 

OPERATION. 

£.    s. 
I  Is.  4d.  I  yij  I  719  10  price  at  £1 

[per  yd. 

143   18  price  at   4s. 

[per  yd. 

illns.  47  19  4d.  at  \s.  4d. 

[per  yd. 
Here  for  the  sake  of  ease  in  the 
operation,  because  5x3=15,  there- 
fore I  divide  the  price  at  one  pound 
\H'.r  yard  by  5,  and  that  quotient  by  3 
wjiich  gives  the  answer. 


DOLLARS,   CENTS,   HILLS. 


Is. 


•.  What  will  648  yards   cost   at 
Qd.  per  yard  ?  Ans.  £54. 


3.  What  will   1671  yards  cost  at 
3s.  4d.  per  yard  ?     .^ns.£27  18s.  4d. 


13.  What  will  7191  yards  cost  at 
22  cents,  3  mills  per  yard  ? 

Ans.  $160,448. 


14.  What  will  648  yards  cost  at 
27  cents  and  8  mills  per  yard  '? 

Ans.  g' 180, 144. 


15.  What  will  1671  yards  cost  at 
:5  cents,  6  mills  per  yard  ? 

Ans.  $93,13. 


148 


PRACTICE. 


Sect.  II.  10. 


PoLxns,  sniLr.iNGS,  rcNcE,  faktu. 

4.  What  will  6871  yards  coa-t  at 
6j.  per  yard  ?     Jlns.  £lll  Ms.'  6d. 


CASE  4. 

When  the  price  of  \yd.  4'C.  is  shil- 
linai),  ur  shillings,  pence  and  farthings, 
iLid  noi  an  even  part  of  £,1.  Multi- 
ply tlie  value  of  the  quanlity  at  Is. 
)>€r  yard  by  the  number  of  shillings  ; 
f<»r  the  pence  and  tartiiint^s,  take 
!» trts,  as  in  case  2,  the  resuUs  added 
will  i;ive  the  answer,  which  must  be 
rediued  to  pounds. 

//"  (he  price  be  shillings  only,  and  an 
even  nuinber ;  multiply  by  half  the 
price  or  even  number  of  shillings  for 
one  yard,  double  the  unit  figure  of 
the  product  for  shiUings,  the  remain- 
ing figure  will  be  pounds. 

Note.     When  the  quantity  con- 
triins  a  fraction,   work  for  the  inte- 
gers, and  for  the  fraction  take  pro- 
portional jfarts  of  the  rate. 
EXAMPLES. 

1.  What  will  167^  yards  cost  at 
1 7  J.  ijd.  per  yard  ? 

OPEKATION. 

s. 

I  i^d.  I  i  I  167 
17 


1169 
167 


2839.pncc  at  Ms.  per  yd. 
83  6 — at  6d.  per  yard. 
8  9  price  of  3.  yard. 

2|0)5>93|]  3d. 

■Int.  £i4ii  I  l.v  3d. 


DOLLARS,    CENTS,    MILLS. 

16.  What  will  687i  yards  cost  at 
83  cents,  3  mills  per  yard  ? 

Ans.  ^572,687. 


17.  What  will  167|  yards  cost  at 
$2,916?  ^7j*.  j^4 88,43. 


Sect.  il.  10. 


PRACTICE. 


149 


POi-NDS,  SHILLINGS,  PENCE,  FARTH. 

2.  What  will  5482  yards  cost  iit 
1 2s.  4id.  per  yard  ? 

Ans.  £3391    IPs.  9d. 


3.  What  vUI   G14  yards   cost   at 
16s.  per  yard  ? 

OPERATION. 
G14 

8  half  the  price. 

4012  double  the  first  figure 
£491   4s.  Ans.         \for  shill. 

4.  What  will    176   yards  cost  at 
12s.  per  yard  1        Am.  £105  12s. 


DOLLARS,    CENTS,   MILLS. 

18.  What  will  5482  yards  cost  at 
^2,063  per  yard  ? 

Ans.  gl  1309,366 


19.  What  will  614  yards  cost  »t 
$2,667  per  yard  ? 

Ans.  $1637,538. 


20.  What  wiU  176  yards  cost  ai 
$2  per  yard  ?  Ai\s.  '^2>b-i. 


5.  What  will  36  yards  cost  at  7s- 
6d.  per  yard?          JJns.  £13  lOs 


21.    What  will    36  yarda  cost  ai 
$1,25  per  yard?  Ans.  %\b. 


150 


PRACTICE. 


Sect.  II.  10. 


TOVXDS,    SHILLINGS,    PENCE,    FARTH. 

CASE  5. 
jyjien  the  price  of  hjd.  lib.  <$•< . 
is  pounds,  shillings  and  pence  ; 
multiply  the  quantity  by  the  pounds, 
and  il"  the  shiUings  and  pence  be  an 
even  part  of  a  pound,  divide  the 
given  quantity  by  that  even  part, 
and  add  the  quotient  to  the  product 
for  the  answer  ;  but  if  they  are  not 
an  even  part  of  £,1,  take  parts  of 
parts  and  add  them  together.  Or, 
you  may  reduce  the  pound  in  the 
price  of  1  yard,  &c.  to  shilhngs.  and 
proceed  as  in  the  case  before. 

EXAMPLES. 
1.    What  will   59   yards  cost,  at 

£6  7s.  6d.  per  yp.rd  ? 

OPERATrON. 

£ 


6s.  is  ^  of  £1. 


59  value  of  £,1  per  yd. 
6 


DOLLARS,    CE.NTS,    MILLS 


354 — at  £6"  per  yd. 
2s.  erf.isiof  5.     14  15s.  oJ  6s.  per  yd. 
7    7  6d.  at  2s.  6d. 
[per  yd. 
Ans.  £376  2s.  Gd.     at  £6 
[7s.  9d. 

3.  What  will   163  yards  cost,  at 
£2  8s.  per  yard?     ^ns.  £391  4j. 


22.  What  will  59  yards  cost,  at 
^21,25  per  yard  ? 

OPERATION. 

D.C. 

21,25 
69 


191  25 

1062  5 


gl253  75  .^TW. 

23.  What  will  163  yards  cost,  at 
3  per  yard?  ^ns.  $1304.   , 


Sect.  II.  10. 


PRACTICE. 


151 


ro'JNDS,  SHILLINGS,  PENCE,  FARTH. 

J.  What  will  76  yards  cost  at  £3 
'2s.  Id.  per  yard  ? 

OPERATION. 
S. 

6d.  is  i  of  Is.    78  value  at  Is.  per  yd. 
(j2=shillings  in  £3  2s. 


1 52  value  at  2s.  per  yd. 
456  — at  60s.  per  yd. 
1  d.  i?  i  of  6rf.    38 — at  6d.  per  yard. 
6  4d. — at  Id.  per  yd. 

2|0)475|6 


Ans.  £237  16s.  4d. 

4.  What  is  the  value  of  84  yards 
at  £2  1  Is.  per  yard  ? 

^7iy.£226  16s. 


DOLLARS,    CENTS,    MILLS. 

24.  What  will  76   yards  cost  at 
^10,43  per  yard? 

Ane.  $792,68. 


25.  What  is  the  value  of  84  yards 
at  ^9  per  yard  ?  Ans.  g766. 


152  SLTPLEMBNT  TO  PRACTICE.  Sect.  H.  10. 

SUPPLEMENT  TO  PRACTICE, 

QUESTIONS. 

1.  What  is  Practice  ? 

2.  Why  ia  it  so  called  ? 

3.  When  the  price  of  1  ^'anl,  &c.  is  farthings,  how  is  the  value  of  any 

given  quantity  found  at  the  same  rate  ? 

4.  When  the  price  consists  of  pence  and  farthings,  and  is  an  even  part 

of  Is.  how  is  the  value  of  any  i^iven  quantity  found? 
fi.  When  the  price  is  pence  and  farthings  and  not  an  even  part  of  Is. 
what  is  the  method  of  procedure  ? 

6.  When  the  price  consists  of  shillings,  pence  and  farthings,  how  is  Ihr- 

value  of  any  given  quaniity  found  ? 

7.  When  the  price  contains  shillings  and  pence  and  an  even  part  of  Xi, 

how  is  the  operation  to  be  conducted? 

8.  When  the  price  consists  of  shillings  only,  and  an  even  number,  wh^t 

is  the  most  direct  way  to  find  the  value  of  any  given  quantity  ? 

9.  When  the  quantity  contains  fraction;,  as  \,  },  £,  &,c.  how  are  they  b> 

be  treated  ? 

10.  When  the  price  consists  of  pounds,  and  lower  denominations,  how  ii 

the  value  of  any  given  quantity  found  ? 

11.  When  the  prices  are  given  in  dollars,  cents,  and  mills,  how  is  the 

value  of  any  given  quantity  found  in  Federal  Money  ? 

12.  What  is  the  method  of  proof? 

13.  How  are  operations  in  Federal  Money  proved  ? 

EXERCISES  /JV  PRACTICE. 

In  the  following  exercises  the  attention  of  the  scholar  must  be  excited 
first  to  consider  to  which  of  the  preceding  cases  each  question  is  to  be  re- 
ferred. That  being  ascertained,  he  will  proceed  in  the  operation  accord- 
ing to  the  instruction  there  given. 

1.  What  will  7453 yards  cost  at  lit/,  per  yard  ?  Ans.  £31  3s.  l^d. 

Under  which  of  the 
preceding  cases  does 
this  question  properly 
belong  ? 

Wiiat  must  be  done 
with  the  fraction  (|  of  a 
yard)  in  the  quantity  ? 


i^ECT.  II.  10.  SUPPLEMENT  TO  PRACTICE.  133 

\2.  What  will  964  yards  cost  at  Is.  Bd.  per  yard  ?      Ans.  £80  65,  Bd. 


OFBRATION 


FROOF. 


3.  What  will  354^  yards  cost,  at         4.  What  will  316  yards  cost,  at 
^d.  per  yard  ?  Ms.  Is.  4^d.         ^d.  per  yard  ?  Am.  \9s.  9d. 


5.  Whjt  will  667i  yards  cost,  at 
^id.  per  yird  ? 

Ans.  £3  10s.  llirf. 


6.  What  will  9131  yards  cost,  at 
Gd.  per  yard  ? 

Ans.  £22  16s.  &d. 


U 


154  SUPPLEMENT  TO  PRACTICE.  Sect.  II.  10. 

7.  What  will  912i  yards  cost,  at         8.  What  will  76  yards  cost,  at  i?(/ 
9d.  per  yard  ?  per  yard  ? 

Ans.  £34  4s.  Ud.  Ans.  12s.  Qd 


i 


9.  What  will  845  yards  cost,  at  8s.         10.  What  will  91  yards  come  to 
per  yard  1  at  16s.  per  yard  ? 

Jlns.  £338.  Ans.  £72  16s. 


11.  What  will  1561  yards  come  to,         12.   What  will  96  yari  cost  at 
at  6s.  Ad.  per  yard?  10s.  \\d.  per  yard  ? 

Ans.  £49  lis.  2d.  Ans.  £.8  12s. 


Sect.  II.  10.  SUPPLEMENT  TO  PRACTICE.  155 

13.  What  will  671  yards  cost,  at         14.  What  will  843  yards  cost,  at 
!  12*.  2d.  per  yard  ?    Jlns.  £41  Is.  3d.     6».  8d.  per  yard  ?         Ans.  £281. 


'     15.  What  will  75  yards  cost,  at         16.  What  will  59  yards  come  to, 
£3  3s.  4d.  per  yard  ?  at  £6  7s.  6d.  per  yard  ? 

Ans.  £237  10s.  Ans.  £376  2j.  6d. 


17.  What  m\l  59|  yards  come  to,         18.    What  will  68  yards  cost,  at 
at  £3  6s.  8c/.  per  yard  ?  £4  6s.  per  yard  ? 

Ans.  £199  3s.  4(/.  Ans.  £292  8s. 


ISe  SUPPLExAlENT  TO  PRACTICE.  Sect.  II.  10. 

N.  B.  The  following  questions  are  left  without  any  aaswers,  that  the 
Scholar  may  operate  and  prove  each  question. 

19.  What  will  1 1  yards  of  flannel,  at  2s.  6d.  per  yard  come  to  ? 

OPERATION.  PROOF 


20.  What  will  131b.  of  cotton  cost  at  3s.  4c?.  per  lb.  ? 


21.  What  will  183  yards  of  ribbon  come  to  at  Sd.  per  yard  ? 


THE 

SCHOLARS  ARITHMETIC. 


SECTION  III. 

lULES  OCCASIONALLY  USEFUL  TO  MEN  IN  PARTICULAR  CALLINGS  AND 
PURSUITS  OF  LIFE. 


♦  1.  INVOLUTION. 

Involution,  or  the  raising  of  powers,  is  the  multiplying  of  any  giren 
number  into  itself  continually,  a  certain  number  of  times.  The  quantities 
in  this  way  produced,  are  called  powers  of  the  given  number.     Thus, 

4x4=  16  is  the  second  power  or  square  of  4.  =4' 

4x4x4=  64  is  the  3d  power,  or  cube  of  4.  =4' 

4X4X4X4=266  is  the  4th  power  or  biquadrate  of  4.  =4* 

The  given  number,  (4)  is  called  the  first  power  ;  and  the  small  figure, 
which  points  out  the  order  of  the  power,  is  called  the  Index  or  the  Ex- 
ponent. 


J  2.  EVOLUTION. 


Evolution,  or  the  extraction  of  roots,  is  the  operation  by  which  we 
find  any  root  of  any  given  number. 

The  root  is  a  number  whose  continual  inuUiplication  into  itself  pro- 
duces the  power,  and  is  denominated  the  square,  cube,  biquadrate,  or  2d, 
3d,  4th,  root,  &c.  accordinely  as  it  i«,  %vheu  raised  to  the  2d,  3d,  4th,  &c. 
power,  equal  to  that  power.  Thus,  4  is  the  square  root  of  16,  because 
4X4=16.  4  also  is  the  cube  root  of  64,  because  4x4x4=64  ;  and  3  is 
the  square  root  of  9,  and  12  is  the  square  root  of  144,  auJ  the  cube  root  of 
17-28.  because  12xl2xl"=«IT'J8,  and  no  ou. 


158  EXTRACTION  OF  THE  SQUARE  ROOT.       Sect.  III.  3, 

To  every  number  there  is  a  root,  although  there  are  numbers,  the  pr« 
eise  roots  of  wliich  can  never  be  obtained.    But  by  the  help  of  decimals,  ue 
can  approximate  towards  thofe  root*,  to  any  necessar}  degree  of  exjictness. 
Such  roots  are  called  Surd  Roots,  in  distinction  from  those  perfectly  accu- 
rate, which  are  called  Rational  Roots. 

'i'he  square  root  is  denoted  by  this  character  •%/  placed  before  the  power  ; 
the  other  roots  by  the  same  character,  with  the  index  of  the  root  placeil 
over  it.     Thus  the  square  root  of  16  is  expressed  V*  1*^5  and  the  cube  root 

of  27  is  a/  27,  &c. 

When  the  power  is  expressed  by  several  numbers,  with  the  sign  +  or— 
between  them,  a  line  is  drawn  from  the  top  of  the  sign  over  all  the  parts  of 
it;  thus   tlie  second   power  of  21 — 5  is  \/  21 — 5,  and  the  3d  power  of 

66-f8  is  V5G-f#,  &c. 

The  second,  third,  fourth  and  fifth  powers  of  the  nine  digits  maybe  seen 
in  the  Ibllowing 

T.WLE. 


Boots  ...  - 

1  or  1st  Powers 

1 

^ 

3 

4 

5  1         6  1 

7 

8 

9 

Sfiuarcs  -  -  - 

1  or  i-'d  Towers 

1 

4 

i'l 

16 

25  1      36  1 

49 

64 

bl 

Cubes  -  -  -  - 

1  or  8d  I'owers 

«l 

27  1 

64 

125  j    216  ! 

343 

■512 

729 

Bi<]iiudratcs 

1  or4tli  Powers 

1 

16 

81 

25(> 

625  1  ]2y6  1 

2401 

4096 

6561 

Sursolids  -  - 

1  orotli  Powers 

1 

■S2 

243 

1024 

3125  1  7776  j 

16807 

3276S 

5yu4a 

^  3.  EXTRACTION  OF  THE  SQUARE  ROOT,  f 

A 

To  extract  the  square  root  of  any  number,  is  to  find  another  numbej' 
which  multiplied  by  or  into  itself,  would  produce  the  given  number  ;  and 
after  the  root  is  fouud,  such  a  multiphcaiion  is  a  proof  of  the  work. 

RULE. 

1.  "  Distinguish  the  given  number  into  periods  of  two  figures  each,  by 
putting  a  point  over  the  ])lace  of  units,  another  over  the  jdace  of  hunchcds, 
and  so  on,  which  points  shew  the  number  of  figures  the  root  will  consist  ofi 

2.  "  Find  the  greatest  square  number  in  the  first,  or  left  hand  period^ 
place  the  root  of  it  at(the  right  hand  of  the  given  number,  (after  the  m;in- 
ner  of  a  quotient  in  division)  for  the  first  figure  of  the  root,  and  the  squiire 
number,  under  the  period,  and  subtract  it  therefrom,  and  to  the  remainder i 
bring  down  the  next  period  for  a  dividend. 

3.  "  Place  the  double  of  the  root,  already  found,  on  the  left  hand  of  the 
dividend  for  a  divisor. 

4.  "  Seek  how  often  the  divisor  is  contained  in  the  dividend,  (except  the 
right  hand  ligure)  and  place  the  answer  in  the  root  for  the  second  tigurc  of 
it,  and  likewise  on  the  right  hand  of  the  divisor  ;  multijdy  the  divisor  with 
the  fi;.'ure  last  annexed  liy,ilie  .figure  last  })lared  in  the  root,  and  subtract 
the  j<roduct  lion)  the  dividend  ;  to  the  remauider  join   Uie  next  puiiod  for 

^  new  dividend.  , 


Sect.  III.  3.     EXTRACTION  OF  THE  SQUARE  ROOT.  15^ 

5.  "  Double  the  fisinres  already  found  in  the  root  for  a  new  divisor,  for 
bring  down  your  hist  divisor  for  a  new  one,  doubling  the  right  hand  figure 
of  it)  and  from  these  find  the  next  tigure  in  the  root,  as  last  directed,  and 
continue  the  operation  in  the  same  manner  till  j'ou  have  brought  down  all 
the  periods." 

"  Note  1.  If,  when  the  given  power  is  pointed  off  as  the  power  re- 
quires, the  left  hand  period  should  be  deficient,  it  must  nevertheless  stand 
as  the  first  period." 

"  Note  2.  If  there  be  decimals  in  the  given  number  it  must  be  pointed 
both  ways  from  the  place  of  units  ;  If,  when  there  are  integers,  tihe  first 
period  in  the  decimtds  be  deficient,  it  may  be  completed  by  annexing  so 
many  cyphers  as  the  power  requires  :  And  the  root  must  be  made  to  con- 
sist of  so  many  whole  numbers  and  decimals  as  there  are  periods  belongiiig 
to  each  ;  and  when  the  periods  belonging  to  the  given  niithber  are  exhausted, 
the  operation  may  be  continued  at  pleasure  by  annexing  cyphers." 
■,f  EXAMPLES. 

1.  What  is  the  square  root  of  729  1 

OPERATION. 

729(27  the  root.  The  given  number  being  distinguished  into 

4  periods,  1  seek  the  greatest  square  number  in 

the  left  hand  period  (7)  which  is  4,  of  which 

47)329  the  root  (2)  being  placed  to  the  right  hand  of 

329  the  given  number,  after  the  manner  of  a  quo- 

tient,  and  the  square  number  (4)  subtracted 

000  from  the  period  (7)   to  the  remainder  (3)  I 

bring  down  the  next  period  (29)  making  for 

PROOF.  a   dividend,   329.     Then    the  double  of  the 

27  root  (4)  being  placed  to  t4ie  left  hand  for  a 

27  divisor,   I  say  how  often  4  in  32  ?  {excepting 

~—  9  the  right  handjigurc)  the  answer  is  7,  which 

189  I  place  in  the  root  for  the  second  figure  of  it, 

64  and  also  to  the  right  hand  of  the  divisor  ;  then 

'  multiplying  the  divisor  thus  increased  by  the 

729  figure  (7)  last  obtained  in  the  root,  I  place  the 

product  underneath  the  dividend,  and  subtract  it  therefrom,  and  the  work 

is  done. 

DEMONSTRATION 
Of  the  reason  and  nature  of  the  various  steps  in  the  extraction  of  the  Square 

Root. 
The  superficial  content  of  any  thing,  that  is  the  number  of  square  feet, 
yards  or  inches,  &c.  contained  in  the  surface  of  a  thing,  as  of  a  table  or  floor, 
a  picture,  a  field,  kc.  is  found  by  multiplying  the  length  into  (he  breadth. 
Jf  the  length  and  breadth  be  equal,  it  is  a  square,  then  the  measure  of  one 
of  the  sides  as  of  a  room,  is  Ihe  root,  of  which  tb.e  superficial  content  in  the. 
floor  of  that  room  is  the  second  power.  So  that  having  the  superficial 
contents  of  the  floor  of  a  square  room,  if  we  extract  the  square  root,  we 
shall  have  the  length  of  one  side  of  that  room.  On  the  other  hand,  having 
the  length  of  one  side  of  a  square  room,  if  we  muiiiply  Ihnt  number  into 
itself,  that  is,  raise  it  4o  the  second  pywer,  we  shall  tii&n  have  the  super- 
llcial  contents  of  the  floor  of  that  room. 

The  extraction  of  the  square  root  therefore  lias  tl.is  operrdion  on  nuin- 
bcr«.  to  orrcHge   the  munlcrs   of  zvhich  U'e   extvLtCt   i\c   root  into  ■■ 


160 


EXTRACTION  OF  THE  SQUARE  ROOT.       Sect.  III.  3 


G26(20 
4 


ri.>.  1. 


form.  .As  if  :•-  m.in  slujtild  have  625  yards  of  carpeting  1  yard  v.ide,  if  lie 
extract  U>e  t<^ufire  root  oi  that  rm!nl)er  (625)  lie  tvill  then  b^ive  flie  IcngUi 
o(  uiic  si^k-  iA  u  sijUHi-e  room,  llie  floor  uf  ^vLicb,62o  yards  will  be  just 
b;iflicieitt  lo  «;over. 

To  procet-d  (hen  to  the  d:^aion?tralion. 

llx>v!!LE  Si.  .S(ippO!?tv.g  a  iiKtn  hius  G25  yards  of  carpeting,  1  3'ard  wide, 
what  will  be  Hie  lenqlli  oi*  one  side  ol"  a  square  room,  the  floor  of  whit: h 
his  c;!!|)eting  wili  cover. 

The  iirst  step  is  to  j»oint  off  (he  number  into  p<?riods  of  two  figures  each. 
This  determine*  the  n«ii»her  of  tiirures  of  which  tiie  root  will  consist,  uud  is 
f!o;ie  Oil  this  principle,  that  the  product  of  any  tzco  tninibers  can  ftave  at  rnai^t 
f'Ut  so  mavy  places  of  fr^ures  as  there  are  places  in  both  the  factors,  and  ai 
Jcdst  but  one  less,  of  wfiich  any  person  viay  satisfy  himself  at  pleasure. 

OITRjtTION. 

The  number  bein;;  |>ointed  off  as  the  rule 
directs,  we  find  we  have  two  periods,  con- 
soquently  the  root  will  consist  of  two  figures. 
Tlie  greatest  square  number  in  the  left  hand 
period  (G)  is  4,  of  which  two  is  the  root, 
therefore,  2  is  the  first  figure  of  the  root, 
aj)d  as  it  is  certain  we  have  one  figure  more  ' 
to  fini'  in  the  root,  wc  may  for  the  pres^ent 
supply  the  place  of  tl.at  figure  by  a  cypher 
(2t.')  then  20  will  express  the  just  value  of 
that  pnrt  of  the  root  now  obtained.  But  it 
mui^t  be  remembered,  that  a  root  is  the  side 
of  a  square  of  equal  sides.  Let  us  then 
form  a  square.  A,  Fi^.  I.  each  side  of  which 
Bhall  be  supposed  ^0  yards.  Now  the  side  a  b 
of  thi?)  square  or  either  of  the  sides,  shcv.': 
<i  20  b     the  root  20,  which  we  have  obtained. 

To  proceed  then  by  the  rule  "  Place  the  square  nvmber  vuderneath  tur. 
period,  subtract,  and  to  the  remainder  bring  doxvn  the  next  period.^^ — Now  the 
ftquare  number  (4)  is  the  superficial  contents  of  the  square  A  made  evident 
thus — each  side  of  the  square  A  measures  20  yards,  which  number  multipli- 
ed into  itself,  produces  400,  the  superficial  contents  of  the  square  A.  al-^o  tlit. 
square  number,  or  the  square  of  the  figure  2  already  found  in  the  root,  is  4, 
vv!sic1i  placed  under  the  j)eriod  (6)  as  it  falls  in  the  j)lace  of  hundreds,  i->  in  i«  - 
ality  400,  as  might  be  seen  also  by  filling  the  pl;;ces  to  the  right  hand  with  f \\- 
phers,  then  1  subtracted  from  6  and  to  the  remainder  (2)  the  next  period  (2  ) 
beiiiji;  brought  down,  it  is  plain,  the  sum  G25  has  been  dimini-hed  by  the  u- 
di'.ction  of  sOO,  a  number  equal  to  tiie  ^uperficial  contents  q'l  the  square  A. 

IJcrice  Fit!.  I.  exiiibils  the  exact  progress  of  the  operation.  By  tie 
operation  400  yards  of  the  carpeting  have  ])een  dispo.sed  of,  and  by  tiie 
iigure  is  seen  the  disposition  made  of  them. 

Now  the  square  A  is  to  be  enLrged  by  the  addition  of  the  225  yards 
which  remaiTi,  and  this  addition  must  be  so  made  that  the  Iigure  yt  the 
fcume  time  shall  conlir.ue  to  be  a  complete  and  perfect  square.  If  the  ad- 
dition be  made  to  one  side  only,  the  figure  would  lose  its  square  form,  it 
nmst  be  made  to  two  sides  ;  for  this  reason  the  rule  directs,  *'  place  the 
eouble  of  the  root  ahetidy  found  on  the  left  hand  of  the  dividend  fo'r  u  di- 
visor." The  double  of  ll.c'  vcv)l  i-  j;ist  equ;-d  to  two  j;ides  b  c  and  c  d  o( 
i'..c  square,  A,  as  may  be  ■^Hksu  by  whA  fidJows. 


Sect.  III.  3.     EXTRACTION  OF  THE  SQUARE  ROOT. 
OPERATION  continued. 


161 


I 


625(25 
4 


45)225 
225 


The  double  of  tbe  root  is  4,  which  placed  for 
a  divisor  in  place  of  tens  {fur  it  must  be  remem- 
bered that  the  next  figure  m  the  root  is  lo  be  placed 
before  it)  is  in  reality  40.  equal  (o  the  sides  b  c 
(20)  and  c  d  (20)  of  the  square  A. 


Fig.  II. 


/S 


20 

6 

c 

6 

D  5 

25 

100 

c 

A 

C 

20 

20 

20 

5 

400 

100 

20 

b      6  h 

=400  yards. 
=100    — 
=100    — 

=  25    — 


Again,  by  l]je>ule,  "  seek  how 
often  tTie  divisor  is  contained  in 
the  dividend  (except  the  right 
hand  figure)  and  place  the  an- 
swer in  the  root,  for  tlic»second 
figure  of  it,  and  on  the  right 
hand  of  the  divisor." 

Now  if  the  sides  b  c  and  c  d  o{ 
the  square  A  Fig.  II.  is  the  length 
tOAvhich  the  remainder  226  yds. 
are  to  be  added,  and  the  divisor 
(4  tens)  is  the  sum  of  these  two 
sides,  it  is  then  evident  that  225 
divided  by  the  length  of  the  two 
sidcs,that  is  by  the  divisor(4  tens) 
will  give  the  breadth  of  this  new 
addition  of  the  225  yards  to  the 
sides  b  c  and  c  d  o£  the  square  A. 

But  we  arc  directed  to  "  ex- 
cept the  right  hand  figure,"  and 
also  to  '''^  place  the  quotient  figure 
on  tJic  right  hand  of  the  divisor  ;*' 
the  reason  of  which  is  that  t'.ie 
addition,  C  ef  and  C  g  h  to  the 
sides  b  c  and  c  d  of  the  square,  A,  do  not  leave  the  figure  a  completp 
square,  but  there  is  a  deficiency  D,  at  the  corner. — Therefore  in  dividi.  ;. 
the  right  hand  figure  is  excepted,  to  leave  something  of  the  dividend,  f^r 
this  deficiency  ;  and  as  the  deficiency  D,  is  limited  by  the  additions  C  cf 
:ind  C g  h,  and  as  the  quotient  figure  (5)  is  the  width  of  these  additic.)?, 
consequently  equal  to  one  side  of  the  square  D  ;  therefore  the  quotient 
figure  (5)  placed  to  the  right  hand  of  the  divisor  (4  tens)  and  multiphed 
into  itself,  gives  the  contents  of  the  square  D,  and  the  4  tcns=to  the  sum 
of  the  sides,  be  and  c  d  of  the  addition  of  Ce/and  Cg  h,  multiplied  by  the 
quotient  figure  (5)  the  width  of  tliose  additions  give  the  contents  C  cy'and 
C  g  h,  which  together  subtracted  from  the  dividend,  and  there  being  no  re- 
mainder, shew  that  the  225  yards  are  disposed  in  these  new  additions  C  cf, 
Cg  h,  and  D,  and  the  figure  is  seen  to  be  continued  a  complete  square. 

Consequently,  Fig.  if.  shews  the  dimensions  of  a  square  room,  25  yardi^ 
on  a  side,  the  floor  of  which  G25  jards  of  carpeting,  1  j-ard  wide  v/ill  be 
sufTicient  to  cover. 

The  Proof  is  seen  by  adding  together  the  different  parts  of  the  figure. 
Such  are  the  principles  on  which  the  operation  cf  extracting  the  square 
root  i=  ffroLindcd. 

W 


Proof  625  yards. 


162  EXTRACTION  OF  THE  SQUARE  ROOT.      Sect.  III.  8» 

3.    What  is  the  square   root  of        4.  What   is  the   square   root  of 
10342656?  Ms.  3216.         43264?  jJns.  208. 


5.  Whil  is  t^ViQaA;  roc*  of  964,5192360241  ?  Ans.  31,05671. 

V 


^acT.  III.  3.    EXTRACTION  OF  THE  SQUARE  ROOT. '  163 

6.   What  is   the   square   root  of        7.    What  is    the  square  root  of 
^98001?  Ms.  999.         234,09?  Ans.  15,3. 


8.  What  is  the  f^dve  root  of  1030892198,4001  i  Am.  32107,51. 


164  tiUPFLEMENT  TO  THE  SqUARE  ROOT.      Sect.  III.  3, 

SUPPLEMENT  TO  THE  SQUARE  ROOT, 


QUESTIONS. 

1.  What  is  to  be  understood  by  a  root  ?  A  power  ?  The  second,  third, 

and  fourth  powers  ? 

2.  V.hat  is  the  Index,  or  Exponent? 

3.  What  is  it  to  extract  the  Square  Root  ? 

A.  Why  is  the  given  sum  pointed  into  periods  of  two  figures  each  ? 

b.  Jn  the  operation,  having  found  the  first  figure  in  the  root,  why  do  wc 

subtract  the  square  number,  that  is,  the  square  of  that  %ure  from 

the  period  in  which  it  was  talccn  ? 

6.  Why  do  we  double  the  root  for  a  divisor  ? 

7.  In  dividing  why  do  we  except  the  right  hand  figure  of  the  dividend  ? 
S.  Why  do  we  place  the  quotient  figure  in  the  root,  and  also  to  the  right 

h;md  of  the  divisor  ? 
#  9.  If  there  be  decimals  in  the  given  number  how  must  it  be  pointed  ? 
10.  How  is  the  operation  of  extracting  the  Square  Root  proved  ? 

EXERCISES  W  THE  SQUARE  ROOT. 

1 .  A  Clergyman's  glebe  consists  of  three  fields  :  the  first  contains  5  acres, 
Qr.  12/).  the  second,  2  acres,  2r.  15p.  the  third,  1  acre,  Ir.  14p.  in  exchange 
for  wliich  the  heritors  agree  to  give  him  a  square  field  equal  to  all  the 
three.     Sought  the  side  of  the  square  ?  Ans.  o9polfs. 


2.  A  general  has  an  army  of  409G 
men  ;  how  many  must  he  place  in  rank 
and  file  to  form  them  into  a  square  ? 
Ans.  64 


Sect.  III.  3.      SUPPLEMENT  TO  THE  SQUARE  ROOT.  I65 

3.  There  is  a  cirde  whose  diameter  is  4  inches,  what  is  the  diameter  ©f 
a  circle  4  times  as  large  ?  Ans.  8  inches. 

Note.  Square  the  given  diameter,  multiply  this 
square  by  the  given  proportion,  and  the  square 
root  of  the  product  will  be  the  diameter  required. 
Do  the  same  in  all  similar  cases. 

If  the  circle  of  the  required  diameter  were  to  be 
less  than  the  circle  of  the  given  diameter,  by  a 
certain  proportion,  then  the  square  of  the  given 
diameter  must  have  been  divided  by  that  pro- 
portion. 


4.  There  are  two  circular  ponds  in  a  gentleman's  pleasure  ground  ;  the 
diameter  of  the  less  is  lOO  feet,  and  the  greiter  is  three  times  as  large. — 
What  is  its  diameter  ?  .^ns.  173,2 -f 


5.  If  the  diameter  of  a  circle  be  12  inches,  what  will  be  the  diameter  of 
another  circle  half  so  large  ?  Ans.  B,4Q-{-inches. 


X9  SUPPLEMENT  TO  THE  SQUARE  ROOT.     Sect.  III.  3 

6.  A  wall  is  36  feet  high,  and  a  ditch  before  it  is  27  feet  wide  ;  what  is 
the  length  of  a  ladder,  that  will  reach  to  the  top  of  the  wall  from  the  oppo- 
site side  of  the  ditch  ?  Jlns.  45  feet. 

Note.  A  Figure  of  three 
sides,  like  that  formed  by 
the  wall,  the  ditch  and  the 
ladder,  is  called  a  right  an- 
gled triangle,  of  which  the 
square  of  the  hypothenuse, 
or  slanting  side,  {the  ladder) 
is  equal  to  the  sum  of  the 
squares  of  the  two  other 
sides,  that  is,  the  height  of 
the  wall  and  the  width  of  the 
ditch. 


7.  A  line  of  36  yards  will  exactly  reach  from  the  fi)  of  a  fort  to  the 
opposite  kink  of  a  river,  known  to  be  24  yards  broad  ;  the  height  of  the 
wall  IS  required?  Ms.  20,^3+ yards. 


Sect.  III.  4.     EXTRACTION  OF  THE  CUBE  ROOT.  167 

8.  Glasgow  is  44  miles  west  from  Edinbnrgh  ;  Peebles  is  exactty  south 
from  Edinburgh,  and  49  miles  in  a  straight  line  from  Glasgow  ;  what  is  the 
distance  between  Edinburgh  and  Peebles  ?  .ins.  2\y5-{-miUi. 


^  4.  EXTRACTION  OF  THE  CUBE  ROOT. 


To  extract  the  Cube  Root  of  any  number  is  to  find  another  number, 
which  multiplied  into  its  square  shall  produce  the  given  number. 

RULE. 

1.  "  Separate  the  given  number  into  periods  of  three  figures  each,  by 
piitlinga  point  over  the  unit  figure,  and  every  third  figure  beyond  the  place 
of  units. 

2.  "  Find  the  greatest  cube  in  the  left  hand  period,  and  put  its  root  rn 
tlie  quoliont. 

3.  "  Subtract  the  cube  thus  found,  from  the  said  period,  and  to  the  re- 
mainder brin,^  down  the  next  period,  and  call  this  the  dividend. 

4.  "  Multiply  the  square  of  the  quotient  by  300,  calling  it  the  triple 
square,  and  the  quotient  by  30,  calling  it  the  triple  quotient,  and  the  sum 
of  these  call  the  divisor. 

5.  "  Seek  how  often  the  divisor  may  be  had  in  the  dividend,  and  place 
the  result  in  the  quotient. 

G.  "  Multiply  the  tiiple  square  by  the  last  quotient  figure,  and  write  tlie 
product  under  the  dividend  ;  multiply  the  square  of  the  last  quotient  figure 
by  tlie  triple  quotient,  and  place  this  product  under  the  last ;  under  all, 
set  the  cube  of  the  last  quotient  figure,  and  call  their  sum  the  subtrahend. 

7.  "  Subtract  the  subtrahend  from  the  dividend,  and  to  the  remainder 
iiring  down  the  next  period  for  a  new  dividend,  with  which  proceed  as  be- 
fore, and  so  on  till  the  whole  be  finished. 

NoTK.  "The  same  rule  must  be  observed  for  continuing  the  operation, 
^nd  pointing  for  decimals,  as  in  <jic  square  ro.^t." 


168 


EXTRACTION  OF  THE  CUBE  ROOT.      Sect.  III.  1. 


1.  What  is  the  cube  root  of  373248  ? 

OrERATIO.V. 


343 


:18(72  the  root. 


7X7X300=14700,  the  triple  square. 
7x30        =     210  the  triple  quotifiit 

14910  the  divisor. 
14700X2=^29400 
2x2x210=     U40 
2X2X2     =        8 


30248  the  subtrahend. 


Divisor  14910)30243 

29400 

840 

8 

30218 
OOUUO 

DEMOXSTR,iTIO\ 

Of  the  reason  arid  nature  of  the  various  steps  in  the  operation  of  extracting 

the  Cube  Root. 

Any  solid  body  having;  six  equal  sides,  and  each  of  the  sides  an  exact 
square  is  a  Cuee,  and  the  mfasure  in  length  of  one  of  its  sides  is  the  root 
ol  that  cube.  For  if  the  measure  in  feet  of  any  one  side  of  such  a  body  be 
multiphed  three  times  into  itself,  that  is,  raised  to  the  third  pov.-er,  the 
product  will  be  the  number  of  solid  feet  the  whole  body  contuins. 

And  on  the  other  hand,  if  the  cube  root  of  any  number  of  feet  be  ex- 
tracted, this  root  will  be  the  length  of  orj^  side  of  a  cubic  body,  the  whole 
contents  of  which  will  be  equal  to  such  a  number  of  feet. 

S;jpposing  a  man  has  13824  feet  of  timber,  in  distinct  and  separate  blocks  of 
one  toot  each  ;  he  wishes  to  know  how  large  a  solid  body  they  will  make  when 
laid  together,  or  what  will  be  the  length  of  one  of  the  sides  of  that  cubic  bod}  ? 

To  know  this,  all  that  is  necessary  is  to  extract  the  cube  root  of  that 
number,  in  doing  which  I  propose  to  illustrate  the  operation. 

OPERATION. 

In  this  number,  pointed  off  as  the  rule  directs, 
13824(20  tliere  are  two  periods  :  of  course  there  will  be 

8  two  figures  in  the  root. 

The  greatest  cube  in  the  right  hand  period 
(13)  is  8,  of  which  2  is  the  root,  therefore  2 
placed  in  the  quotient  is  the  first  figure  of  the 
root,  and  as  it  is  certain  ive  have  one  figure  more 
to  find  in  the  root,  we  may  for  the  present  sup- 
pl}'  the  place  of  that  one  figure  by  a  cypher  (20) 
then  20  will  express  the  true  value  of  that  part 
of  the  root  now  obtained.  But  it  must  be  re- 
membered, that  the  cube  root  is  the  length  of 
one  of  the  sides  of  the  cubic  body,  whose  length, 
breadth  and  thickness  are  equal.  Let  us  then 
form  a  cube.  Fig.  1,  each  side  of  which  shall  be 
supposed  20  feet ;  now  the  side  A  B  of  this  cube, 
or  either  of  the  sides,  shews  the  root  (20)  which 
we  have  obtained. 


feet={he  soliil  contents  of  the  Culc. 


Sect.  III.  4.       EXTRACTION  OF  THE  CUBE  ROOT.  169 

The  Rule  next  directs,  subtract  the  cube  thus  found  from  the  said  period, 
und  to  the  remainder  bring  doinn  the  next  period,  4'c.  Now  this  cube  (8)  is 
the  gohd  contents  of  tha  figure  we  Have  in  representation.  Made  evident 
thus — Each  side  of  this  figure  is  20,  which  being  raised  to  the  3d  power, 
that  is  the  length,  breadth  and  thickness  being  muUipUed  into  each  other, 
gives  tt^  solid  contents  of  that  figure^="8000  feet,  and  the  cube  of  the  root, 
(iJ)  which  we  have  obtained,  is  8,  vvhich  placed  under  the  period  from 
nhich  it  was  taken  as  it  falls  in  the  place  of  thousands,  is  8000,  equal  to  the 

lid  contents  of  the  cube  A  B  C  D  E  F,  which  being  subtracted  from  the 

.en  number  of  feet,  leaves  5824  feet. 

Hence,  Fig.  I.  exhibits  the  exact  progress  of  the  operation.  By  the  ope- 
:  ition  8000 feet  of  the  timber  are  disposed  of,  and  the  figure  shews  the  dis- 
position made  of  them,  into  a  solid  pile,  which  measures  20  feet  on  every  side. 

Now  this  figure  or  pile  is  to  be  enlarged  by  the  addition  of  the  5824  feet, 
which  remains,  and  this  addition  must  be  so  made,  that  the  figure  or  pile  shall 
continue  to  be  a  complete  cube,  that  is,  have  the  measure  of  all  its  sides  equal. 

To  do  this  the  addition  must  be  made  equally  to  the  three  difierent 
6;quares,  or  faces  a,  c  and  b. 

The  next  step  in  the  operation  is,  to  find  a  divisor ;  and  the  proper  di- 

or  will  be,  the  number  of  square  foet  contained  in  all  the  points  of  the 

:  ire,  to  which  the  addition  of  the  5824  feet  is  to  be  made. 

ilcnce  we  are  directed  to  "  multiply  the  square  of  the  quotient  by  300,' 

ihe  object  of  which  is  to  find  the  superficial  contents  of  three  faces  a,  c,  b, 

;o  which  the  addition  is  now  to  be  made.     And  that  the  square  of  the  quo- 

i.ent,  multiphed  by  300  gives  the  superficial  contents  of  the  faces  a,  c,  b, 

!.->  evident  from  what  follows  : 

Side  A  B=20  \  2  quotient  figure. 

Side  A  F=20 (    r,,    r  2 
>  oj  the  jace  a. 

Superficial  coTiients'=^'iOO  )  4  the  square  of  5. 

3  300 


The  triple  square  l'zOO=the         The  triple  square   1200=the   superficial 
.'i}rj)prjicial  contents  of  the  faces     contents  of  the  faces  a,  c,  and  b. 
a,  '-,  andb.  Here  the  quotient  figure  2  is  properly, 

1'he  two  sides  A  B  fc  A  F  txco  tens,  for  there  is  another  figure  to  fol- 
of  the  fdcc  a,  multiplied  into  low  it  in  the  root,  and  the  square  of  2,_ 
each  other,  give  the  superfi-  standing  as  units,  i.s  4,  but  its  true  value 
l^cial  content  of  c,  and  as  the  is  20  (^the  side  A  B)  of  which  the  square 
faces,  a,  c,  and  6,  are  all  equal,  is  400,  we  therefore  lose  two  cyphers,  and 
therefi^rc  the  content  of  iUcc  these  two  cyphers  are  annexed  to  the  fig- 
0,  multiplied  by  3,  will  give  ure  3 — Hence  it  appears  that  we  square 
the  coatenls  of*;,  c,  and  h.  the  quotient  with  a  view  to  find  the  super- 

ficial content  of  the  face  or  square  a,  we 
multiply  the  square  of  the  quovient  by  3,  to  find  the  suparRcial  contents  of 
tiie  three  squares,  c,  c,  and  b,  and  two  cyphers  are  annexed  to  the  3,  because 
in  the  square  of  the  quotient  tr^o  cyphers  were  lost,  the  quotient  requiring 
a  cypher  before  it  in  order  to  express  its  true  v.«»  jc,  which  would  throw  the 
quotient  ('!)  into  the  place  of  tens,  whereas  now  it  stands  in  the  place  of  units. 

Now  wlu  n  tlie  additions  are  made  to  the  squares  a,  c,  and  h,  there  will 
f'videiitiy  bo  a  deficiency,  along  the  v/liole  length  of  the  sides  of  the  squares, 
Isctvveen  each  of  the  additions,  which  must  be  supplied  before  the  figure 
can  hf.  a  complete  cube.  These  dcncicncics  will  be  3,  as  may  be  seen, 
It'i^r.  ll.  nun.  X 


170 


EXTRACTION  OF  THE  CUBE  ROOT.      Sect.  III.  4. 


Therefore  it  is,  that  we  are  directed,  "  mvUipIy  the  quotient  by  30,  calling 
it  the  triple  quotient.^ ^ 

The  triple  quotient  is  the  sum  of  the  three  hues,  or  sides,  agninst  which 
«rc  the  deGcitiicii;s  n  n  n,  all  which  meet  at  a  point,  nigh  the  centre  of  the 
fjgure.     This  is  evident  froai  what  follov.'s. 

The  deficiencies  are  three  in  number, 
they  are  the  ^vhole  length  of  the  sides,  2  quotient. 

the  length  of  each  .'ide  u  20  feet,  oO 

Therefore  'i(i  — 

3 


Triple  quotient  60=^o  the  length 
of  3  sides  ■where  are  dejiciencics  to 
be  jilled. 


Triple  quotient  60  equal  tlw.  IcngtJi 
of  3  sides,  fyc. 
Here  as  before,  the  quotient 
lacks  a  cypher  to  the  ri^ht  hand, 
to    exhibit  its   true    value  ;  the 
quotient  itsnlf  is    the  length   of 
one  of  the  sides,  whore  are  the  deficiencies  ;  it  is  midtiplied  bj'  3  becaujo 
there   are  3  deficiencies,  and  a  cypher  is  annexed  to  the  3  because  it  has 
I'cen  omitted  in  the  quotient,  which  gives  the  same  product,  as  if  the  true 
value  of  the  quotient  20,  had  been  multiplied  by  3  alone. 
1200  the  triple  square. 
We  now  have  ^      60  the  triple  quotient. 


The  sum  of  which,  1260  is  the  divisor,  equal  the  number  of  square 
feet  contained  in  all  the  points  of  the  figure  or  pile,  to  which  the  addition 
of  the  5824  feet  is  to  be  matle. 


OPERATION 

138 


continued. 


1(24  the  root. 


Divis.  1260)o82'l  the  dividend. 
"4800" 
960 
64 


A       20        F 

1200  triple  square. 

4  last  quotient  Jtgure. 


This  Figure  in  the  root  (4)  shews 
tl)e  depth  of  the  addition  on  every  point 
where  it  is  to  be  made  to  the  pile  or 
figure,  reproeented,  Fig.  I. 


Frc,  II.  exhibits  the  additions  made  to 
the  squares  a  c  h,  by  which  they  are 
covered  or  raised  by  a  depth  of  4  feet. 

The  next  step  in  the  operation  is  to 
find  a  subtrahend,   which  subtrahend  is     j 
the  number  of  solid  feet  contained  in  all  ♦ 
the  additions  to  the  cube,  by  the  last 
figure  4. 

Therefore  the  rule  directs,  multiply 
the  triple  square  by  the  last  quotient  Jigure. 

The  triple  square  it  must  be  remem- 
bered, is  the  superficial  contents  of  the 
faces  a  c  and  b,  which  multiplied  by  4, 
the  depth  now  added  to  these  faces,  or 
squares,  gives  the  number  of  solid  feet 
cniitained  in  the  additions  by  the  last 
quotient  figure  4. 


4800  feet, equal  the  addition  trade  to  the  squares,  or  faces,  r.  c,  h,  of  Fig.  I 
a  depth  of  'i  feet  on  each. 


EXTRACTION  OF  THE  CUBE  ROOT. 


171 


o 
60 
16 


F4n 
triple  quotient, 
square  of  the  last  quotient 


Then,  "  Multiply  the  square  of  the  last  quO' 
tient  figure  by  the  triple  quotient."  This  is  to 
fill  the  deficiencies,  n  n  n,  Fig.  II.  Now  these 
deficiencies  are  hmited  in  length  by  the  length 
of  the  sides  (20)  and  the  triple  quotient  is  the 
sum  of  the  length  of  the  deficiencies.  They  arc 
hmited  in  width  by  the  last  quotient  figure  (4) 
the  square  of  which  gives  the  area  or  super- 
ficial contents  atone  end,  which  multiplied  into 
their  length,  or  the  triple  quotient,  which  i« 
the  same  thing,  gives  the  contents  of  tliose  ad- 
ditions 4n4,  4n,  4»,  Fig.  III. 


3G0 
60 


9G0  feet  disposed  in  the  deficiencies,  between  the  additions  to  the  squares 
a  c  b.  Fig.  III.  exhibits  these  deficiencies,  supplied  47)4,  4«,  4n,  and 
discovers  another  deficiency  u-here  these  approach  together,  of  a 
corner  wanting  to  make  the  figure  a  complete  cube. 

Lastly,  "  Cube  the  last  quotient  figure.''''  This 
is  done  to  fill  the  deficiency,  tig.  III.  left  at 
one  corner,  in  filling  up  the  other  deficiencies, 
n  n  n.  This  corner  is  limited  by  those  defi- 
ciencies on  every  side,  which  were  4  feet  in 
breadth,  consequently  the  square  of  4  will  be 
the  solid  content  of  the  corner  which  in 
Fig.  IV.  e  e  e  is  seen  filled. 


Now  the  sum  of  these  additions  make  the 
subtrahend,  which  subtract  from  the  dividend 
and  the  work  is  done. 


16 
4 


64  feet  is  the  cornj-  »  «  c,  n'here  the  additions  n  n  n,  approach  together. 

Figure  IF.  Shews  the  pile  which  13824  solid  blocks  of  one  foot  each, 
would  make  when  laid  together.  Tlie  root  (24)  shews  the  length  of  aside. 
Fig.  I.  shews  tlie  pile  which  would  be  i;>rnied  by  8000  of  Iho.-e  blocks,  fir?t 
hud  together;  Fig.  II.  F'ig.  III.  and  Fig.  T-^  shew  the  changes  which  the 
pile  passes  through  in  the  addition  of  the  re.Tiaining  5824  blocks  or  feet. 

Frnrf.  By  adding  the  contents  of  the  first  figure,  and  the  udditi^ns  ex- 
..bibitcd  in  the  other  fi.'rares  tos-othei'. 


17J  EXTRACTION  OF  THE  CUBE  ROOT.      Stci.  III.  4. 


c- 


Feet. 

8000  Contents  of  Fig.  I. 

4800  addition  to  the  faces  or  square  a,  c,  and  6,  Fig.  U. 
960  addition  to  fill  the  deficiencies  n,  n,  n,  Fig.  ///. 
64  addition  at  the  corner,  c,  c,  c,  FjV.  /r".  where  the  additio 
which  fill  the  deficiencies  n,  n,  n,  approach  together 


4 


13824  Number  of  blocks  or  solid  feet,  all  which  are  now  disposed 

in  Fig.  IV.  forming  a  pile  or  solid  body  of  timber,  24  feei 

on  a  side. 

Such  is  the  demonstration  of  the  reason  ai»d  natureof  the  various  steps  in 

the  operation  of  extracting  the  cube  root.     Proper  views  of  the  figures,  and 

of  those  steps  in  the  operation  illustrated  by  them,  will  not  generally  be 

acquired  without  some  diligence  or  attention.     Scholars  more  especially 

will  meet  with  difficulty.     For  their  assistance,  small   blocks  might  be 

formed  of  wood  in  imitation  of  the  Figures,  with  their  parts  in  diflfcrent 

pieces.     By  the  help  of  these.  Masters,  in  most  instances,  would  be  able 

to  lead  their  pupils  into  the  right  conceptions  of  those  views,  which  are 

here  given  of  the  nature  of  this  operation. 

3.  What  is  the  cube  root  of  2102457G  ?  Ans.  276. 


Sect.  III.  4.      EXTRACTION  OF  THE  CUBE  ROOT.  173 

4.  Wh?t  is  the  cube  root  of  253395799552  ?  Ms.  6328. 


174  EXTRACTION  OF  THE  CUBE  ROOT.     Sect.  III.  4.  I 

5.  What  1*9  the  cube  root  of  84,604519  ?  jj^,.  4  3^. 


6.  What  is  the  cnbe  root  of  2  ? 


JIns.   1,25+ 


Sect.  III.  4.       SUPPLEMENT  TO  THE  CUBE  ROOT.  nr, 

SUPPLEMENT  TO  THE  CUBE  ROOT. 


QUESTIONS. 

1.  What  is  a  Cube? 

2.  What  is  understood  by  tlie  cube  root  ? 
•3.  What  is  it  to  extract  the  cube  root  ? 

4.  In  the  operation,  having  found  the  first  figure  of  the  root,  why  is  the 
cube  of  it  subtracted  from  the  period  in  which  it  was  taken  ? 

6.  Why  is  the  square  of  the  quotient  muUipUed  by  300  ? 
C.  Why  is  the  quotient  mulliphed  by  30  ? 

7.  Why  do  we  add  the  triple  square  and  triple  quotient  together,  and  tlie 

sum  of  them  call  the  divisor  ? 

8.  To  find  the  subtrahend,  why  do  we  multiply  the  triple  square  by  the 

last  quotient  figure  ?  the  square  of  the  last  quotient  figure  by  the 
triple  quotient  ?  Why  do  we  cube  the  quotient  figure  ?  Why  do 
these  sums  added,  make  the  subtrahend  ? 

9.  How  is  the  operation  proved  ? 

EXERCISES  IjY  THE  CUBE  ROOT. 
1 .  If  a  bullet  6  inches  in  diameter  weigh  321b.  what  will  a  bullet  of  the 
same  metal  weigh  whose  diameter  is  3  inches  ?  Jltis.  41b. 

Note.  "  The  solid 
contents  of  similar  fig- 
ures are  in  proportion 
to  each  other,  as  the 
cubes  of  their  similar 
sides,  or  diametei-s." 


176  SUPPLEMENT  TO  THE  CUBE  ROOT.       Sect.  Ill  4 

2.  What  is  the  fide*of  a  cubical  mound,  equal  to  one  288  feet  long,  216 
broad  and  48  feet  high  ?  Ans.   144  feet. 


".  There  is  a  cubical  vessel  \vhose  side  is  two  feet ;  I  demand  the  side 
of  a  vessel  which  shall  contain  three  times  as  much  ? 

Ans.  2  feet  ten  inches  and  ^nearly. 

Note.  Cube  the  given  side, 
multiply  it  by  the  given  pro- 
portion, and  the  cube  root  of 
the  product  will  be  the  siJe 
souffht. 


i 


Sect.  III.  5.  FELLOWSHIP.  177 

*  5.  FELLOWSHIP. 


FELLOWSHIP  is  a  rule  by  which  merchants  and  others,  trading  in 
partnership,  compute  their  particular  shares  of  tUe  gain  or  loss,  in  propor- 
tion to  their  stock  and  the  time  of  it3  continuance  in  trade. 

It  is  of  two  kinds,  Single  and  Double. 

SINGLE  FELLOWSHIP, 

Is  when  the  stocks  are  employed  equal  times. 

^  RULE. 

As  the  whole  sum  of  the  stock  is  to  the  whole  gain  or  loss,  so  is  each 
man's  particular  stock  to  his  particular  share  of  the  gain  or  loss. 

Pkoof.  Add  all  the  shares  of  the  gain  or  los.^  together  ;  and  if  the  work 
be  right,  the  sum  will  be  equal  to  the  whole  gain  or  loss. 

EXAMPLES. 
1.  Two  merchants,  A  and  B,  make  a  joint  stock  of  200  dollars  ;  A  puts 
in  75  dollars,  and  B  125  dollars  ;  they  trade  and  gain  50  dollars  ;  what  is 
each  man's  share  of  the  gain  ? 

OPERATION. 

Dolls.  Dolls.     Dolls, 
As  200  :  60    :  :    75  As  200  :  50  :  :   125 

75  125 

250  250 

350  100 

D.  cts.  50 


200)3750(18,75  A's  share.  ,  D.  cts. 

200  200)6250(3 1 ,25  B's  share. 

600 

1750  

1600  250 


200 

1500  

1400  600 


400 

1000         18,75  A's  share.  

1000         31,25  B's  share.  1000 

1000 


50,00  Proof. 

2.  Divide  the  number^60  into  4  such  parts,  wliich  shall  be  to  each  other 

3,  4,  5,  and  6. 


60  \ 

^ll\  Answer. 
120} 


*'oO  Proof. 


178  SINGLE  FELLOWSHIP.  Sect.  III.  t. 

3.  A  man  died  leaving  3  sons,  to  whom  he  bequeathed  his  estate  in  the 
following  manner,  viz.  to  the  eldest  he  gave  184  dollars,  to  the  second  155 
dollars,  and  to  the  third  96  dollars ;  but  when  his  debts  were  paid,  there 
were  but  184  dollars  left  ;  What  is  each  one's  proportion  of  his  estate  ? 

Ans.  77,829  ) 

65,563  )  Shares*' 
40,606  ) 


4.  A  and  B  cor.panied ;  A  put  in  ,^45,  ami  took  f  of  the  gain  ;  What 
did  B  }.ul  in  ?  Ms.  £30. 


Sect.  III.  5.  DOUBLE  FELLOWSHIP.  179 


DOUBLE  FELLOWSHIP. 

DOUBLE  FELLOWSHIP,  or  Fellowship  with  time,  is  when  the  stocks 
of  partners  are  continued  uiici|uai  times. 

RULE. 

Multiply  each  man's  stock  by  the  time  it  was  continued  in  trade.  Then, 
As  the  whole  sum  of  the  products  is  to  the  whole  gain  or  loss,  so  is  each 
mau's  particular  product  to  his  particular  share  of  the  loss  or  gain. 

EXAMPLES. 
1.  A,  B,  and  C,  entered  into  partnership  ;  A  put  in  85  dollars  for  8 
months  ;  B  put  in  60  dollars  for   10  months  ;  and  C  put  in  120  dollars  for 
[\  months  ;  by  misfortune  they  lost  41  dollars  :   What  must  each  man  sus- 
tain of  the  loss  ? 

OPERATION 

120  680  A's  product. 

3  600  B's  product. 


85 

60 

8 

10 

680 

600 

s  1640  :  41   :  : 

680 

680 

680 

2720 

164|0)2788|0(17  A's  loss. 
164 

1148 

1148 

0000 

As  1640  :  41   : 

:  360 

360 

2460 

123 

164|0)147610 
1476 

^9  C'3  loss. 

360 


360  C's  product. 


1640 
As  1640  :  41   :  :  600 
600 


164jO)2460|0(15  B's  loss. 
164 

820 
820 


Dolls. 
17  A's  loss. 
15  B's  loss. 
y  L'^  1863. 


0000  41   Proof. 


180  DOUBLE  FELLOWSHIP.  Sect.  III.  5. 

2.  A,  B,  and  C,  trade  together  ;  A,  at  first  put  in  480  dollars  for  8 
months,  then  put  in  200  dollai-s  more  and  continued  the  whole  in  trade  8 
months  longer,  at  the  end  of  which  he  took  out  his  whole  stock  ;  B  put 
in  800  dollars  for  9  months,  then  took  out  $583,333  and  continued  the  rest 
in  trade  3  months  ;  C  put  in  ^366,666  for  ten  months,  then  put  in  250  dol- 
lars more,  and  continued  the  whole  in  trade  6  months  longer.  At  the  end 
of  their  partnership  they  had  cleared   1000  dollars  ;  what  is  each  man'^ 

share  of  the  gain  ? 

Ans.  ^378,827  A's  share. 
320,452  B's  share. 
300,721  C's  share. 


SrxT.  III.  5.  SUPPLEMENT  TO  FELLOWSHIP.  181 

SUPPLEMENT  TO  FELLOWSHIP. 

QUESTIONS. 

1.  What  is  Fellowship? 

2.  Of  how  many  kinds  is  Fellowship  ? 

3.  What  is  Single  Fellowship  ? 

4.  What  is  the  rnle  for  operating  in  Single  Fellowship? 
6.  What  is  Double  Fellowship  1 

6.  What  is  the  rule  for  operating  in  Double  Fellowship  ? 

7.  How  is  Fellowship  proved  ? 

EXERCISES  LV  FELLOWSHIP. 

A,  B,  and  C,  hold  a  pastjire  in  common,  for  which  they  pay  £20  per 
annum.  In  this  pasture,  A  had  40  oxen  for  76  days  ;  B  had  36  oxen  for 
TjO  days,  and  C  had  50  oxen  for  90  days.  I  demand  what  part  each  of 
these  tenants  ought  to  pay  of  the  £20. 

£     f.     d.     q. 
Ans.  6     10     2     IffAA  A's  part. 
3     17     1     0|fi§  B's  part. 
9     12     8     2f §j|  C's  part. 


182  BARTER.  Sect.  III.  G     * 

^  6.  BARTER. 

BARTER  is  the  exchanging  of  one  commodity  for  another,  and  tn;t<  hes 
morrhants  so  to  proportion  their  quantities,  that  neither  shall  rnstaiij  lo«6. 
PiiooF.     P>y  changing  llie  order  of  the  question. 

RULE. 

1.  When  the  quantity  nf  (me  commodity  is  givc;i  n:ith  its  value  or  ih^  value  of 
its  integer,  as  also  the  vahiK  of  the  integer  of  some  other  commodity  to  be  <? .•  - 
changed  for  it,  to  find  the  quantity  of  this  commodity  : — Find  the  value  of 
the  commodity  of  which  the  quanlit}'^  is  given,  then  find  hov/  much  of  the 
other  commodity  at  the  rate  proposed  may  be  had  for  that  sum. 

2.  Jf  the  qnavlities  of  both  commoditiea  be  given,  and  it  should  be  required 
to  find  hou-  mi»ch  of  some  other  commodity,  or  how  much  tnoney  should  be  given 
for  the  inequality  of  therr  values  :  Find  the  separate  value  of  the  two  given 
commodities,  subtract  the  l*?ss  froni  the  greater,  and  the  remainder  will  be 
the  balance,  or  value  of  the  other  commodity. 

3.  If  one  commodity  is  rated  above  the  ready  money  price,  iq  flrid  the  bar- 
tr ring  price  of  the  other:  Say,  as  tlie  reiidy  money  price  of  the  one  is  to, 
the  bartering  price,  so  is  that  of  the  other  to  Us  bartering  price. 

EXAMPLES. 

1.  How  much  coffee  at  2^  cents         2.  I  have  760  gallons  of  molasses, 

per  lb.  can   I  have  for  56Ib.    of  tea     at  37  cents  5  mills  per  gallon,  which 

at  43  cents  per  lb.  ?  I   Mould  exchange  for  66  cwt.  2  qr. 

OPERATION.  of  cheese  at  4  dollars  per  cwt.    Must 

5  6  lb.  of  tea.  I  pa}"^  or  receive  money,  and  how 

,4  3  per  lb.  much  ? 

J\ns.  must  receive  ^1^1.   .o 


1 

6 

8 

2  2 

4 

-  lb. 

oz. 

2 

5)2  4, 

0 

8(96 

5^  Am. 

2  2 

5 

8 

1 

5 

1 

5 

0 

8 

2  5)1 

1 

6 

2 

8(5 

1 

2 

5 

Sect.  III.  6.  BARTER.  183 

8.  A  and  B  barter ;  A  has  160  bushels  of  wheat  at  5*.  9rf.  per  bushel, 
for  which  B  gives  65  bushels  of  barley,  worth  2s.  lOd.  per  bushel,  and  the 
balance  in  oats  at  2».  Id.  per  bushel ;  what  quantity  of  oata  must  A  receire 
from  B  ?  -^n*.  325^  bushels. 


4.  A  has  linen  cloth  worth  SOcf.  an  ell,  ready  money;  but  in  barter 
he  would  have  two  shilhogs  ;  B  has  broad  cloth  worth  14s.  6d.  per  yard, 
ready  money ;  at  what  pjice  ought  the  broad  cloth  to  be  rated  in  barter  ? 

Alls.   Ms.  4d.  3<j.-^j per  yard. 


184  SUPPLEMENT  TO  BARTER.  Sect.  111.  6. 

SUPPLEMENT  TO  BARTER. 


QUESTIONS. 

1.  What  is  Barter? 

2.  When  and  how  does  this  rule  become  useful  to  merchants  ? 

3.  When  a  given  quantit}"^  of  one  commodity  is  bartered  for  some  other 

commodity,  how  is  the  quantity  that  will  be  required  of  this  last 
commodity  found  ? 

4.  If  the  quantity  of  both  commodities  be  given,  and  it  be  required  to 

know  how  much  of  some  other  commodity,  or  how  much  money 
must  be  given  for  the  inequahty,  what  is  the  method  of  procedure  ? 

5.  If  one  commodity  be  rated  above  the  money  price,    how  do  you 

proceed  to  find  the  bartering  price  of  the  othsr  commodity  ? 

6.  How  is  Barter  proved  ? 

EXERCISES. 
1.  A  and  B  bartered  ;  A  had  41  cwt.  of  hops,  30s.  per  cwt.  for  which 
B  gave  him  £20  in  money,  and  the  rest  in  prunes,  at  hd.  per  lb.     I  demand 
how  many  prunes  B  gave  A,  besides  the  £20.  Am.  \1C.  3grs.  4/6. 


2.*  How  much  wine  at  gl,28  per  gallon,  must  I  have  for  26cwt.  2qr, 
141b.  of  raisins,  at  $9,444  per  cwt.  ?  Ans.  }96§al.  \qt.  \^t. 


SicT.  III.  7.  LOSS  AND  GAIN.  i»f- 

t  7.  LOSS  AND  GAIN. 


"  Loss  and  Gain  is  a  rule  which  enables  merchant?  to  estinaate  the 
profit  or  loss  in  buying  and  selling  goods  ;  also  to  raise  or  fill  the  price  of 
them,  so  as  to  gain  or  lose  so  much  per  cent." 

CASE  I. 

To  k::.orv  zi>itat  is  gained  or  lost  per  cent.  First,  find  what  the  gain  or 
loss  is  by  subtraction,  then  as  the  price  it  cost  is  to  the  gain  or  loss,  so  is 
^100  (or  JCIOO)  to  the  gain  or  loss  per  cent. 

EXMIPLES. 

1.  If  I  buy  candles  at  16  cents  7         2.  Bought  indigo  at  ^1,20  per  lb. 
mill?  per  lb.  and  sell  them  at  20  cts.     and  sold  the  same  at  90  cents  per  lb. 
per  lb.  what  shall  1  gain  per  cent  or     what  was  lost  per  cent  ? 
iM  hying  out  100  dollars  ?  Ans.  ^25. 

OPERATrON. 

I  sell  at  ,20  per  lb. 
Bought  at  ,167  per  lb. 


I  gain  ,033  per  lb. 

Then  as  ,167   :   ,0  3  3   ::    100 
1  0  0 

D.  cts. 

,167)3,3  0  0(19,76  Ans. 
1  6  7 


J   6  3  0 
15  0  3 

12  7  0 
116  9 


10  10 
10  0  2 


.3.  Bo'ight  37  gallons  of  Brandy  at         4.  Bought  hats  at  4*.  a  piece,  and 

^1,10   per    gallon,    and   sold   it    for  sold  them  again  at  4s.  9 J.  ;  what  is 

,<,'10;  what  was  gained   or  lost  per  the  proHt  in  laying  out  £1^*0  ' 
cent?                     ^ns.  ^1,719  loss.  .lyrs.  £1 8.15s. 


186  LOSS  AND  GAIN.  Sect.  IU.  7. 

CASE  II. 

To  knozv  hpw  a  commodity  must  be  sold  to  gain  or  lose  90  much  per  cent; 
As  100  dollars  (or  £lOO)  is  to  the  price  ;  so  is  100  dollars  (or  £100)  with 
the  profit  added  or  the  loss  subtracted  to  the  gaining  or  losing  price. 

EXAMPLES. 
1.  If  I  bu}'  wheat  at  ^1,25  per  bushel,         2.   If  a  barrel  of  rum  cost 
ijow  must  I  sell  it  to  gain  15  per  cent  ?  15  dollars,  how  must  it  be  sol4 

to  lose  10  per  cent? 

Jlns.  $13,50. 


OPERATION. 

As  100 

1,2 

5  :  :    115 

1    1 

5 

6  2 

5 

1 

2  5 

1 

0 

5 

—  D.  cts.  m. 

y30)i 

4 

3,7 

6(1,43    1  Am 

1 

0 

0 

4 

3  7 

A 

0  0 

3  7 

5 

3  0 

0 

7 

6  0 

7  0  0 

6  0 


3.  If  1301b.  of  steel  cost  £7,  how  mu^t  I  sell  it  per  lb.  to  gain  15^  per 
cent?  Ans.  Is.  ^d.perlh' 


Sect.  III.  7.         SUPPLEMENT  TO  LOSS  AND  GAIN.  187 

SUPPLEMENT  TO  LOSS  AND  GAIN, 


QUESTIONS. 

1.  What  is  Loss  an4  Gain  ? 

2.  Having  the'  price  at  wliich  goods  are  bought  aqd  sold,  how  is  the  loss 

or  gain  estimated  ? 

3.  To  knoTv  )iow  much  a  commodity  must  be  valued  at  to  gaio  or  lose 

so  much  per  cent,  what  is  the  method  of  procedure  ? 

4.  How  may  questions  in  Loss  and  Gain  \te  proved  ? 

EXERCISES. 
1.  A  draper  bought  100  yards  of  broadcloth  for  £56.     I  demand  how 
he  must  sell  it  per  yard  to  gain  £15  in  laying  out  £100  ? 

Ans.  12s.  lOd.  2^. 


2.  Bought  30  hogsheads  of  molasses  at  ^600 ;  paid  in  duties  ^20,66  ; 
for  freight  <^40,78  ;  for  porterage  ^6,05,  and  for  insurance,  $30,84  j  If  I 
eel!  it  at  ^^6  per  hogshead,  how  much  shall  I  gain  per  cent  ? 

.3«s.  g  11,695. 


108  DUODECIMALS.  Sect.  III.  fl. 

$  8.  DUODECIMALS ; 

OR,  CROSS  MULTIPLICATION. 


This  rule  is  particularly  useful  to  Workmen  and  Artificers  in  casting  up 
the  contents  of  their  work. 

Dimensions  are  taken  in  {e^ei,  inchfs  and  part?.  Inches  and  parts  are 
sometimes  called  primes  (')  seconds  (")  thirds  ('")  and  fourths  ("") 

TAIiLE.  By  this  rule  also  may  be  calcula- 

12  Fourths  make  1  Third.  ted  the  solid  contents  of  bodies,  nav- 

12  Tldrds  -  -  -   1  Second.  in<?   the    measures   of  their   different 

1 2  SecniJs  -  -  -   1  Inch  or  prime.      sides,  and  is  very  useful  therefore  in 
12  Inches  or  Pr.    1   Foot.  measaring  wood. 

RULE. 

I.  Under  the  multiplicand  write  the  corresponding  denominations  of  the 
multiplier. 

2  Multiply  each  term  in  the  multiplicand,  beginning  at  the  lowest,  by 
the  feet  in  the  multiplier,  and  write  the  result  of  each  under  its  respective 
tenn,  observing  to  carry  an  unit  for  every  12,  from  each  lower  denomina- 
tion to  its  superior. 

3.  In  the  same  manner  multiply  the  multiplicand,  by  the  inches  in  th;. 
multiplier,  and  write  the  result  of  each  term  in  the  multiplicand,  thus  mul- 
tiplied, 0716  place  to  the  right  hand  in  the  product. 

4.  Proceed  in  the  same  manner  with  the  other  parts  in  the  multiplier, 
which  if  seconds,  write  the  result  izco  places  to  the  right  hand  ;  if  thirds, 
ihrep.  places.  A-c.  and  their  sum  will  be  tlie  answer  required. 

The  more  easily  to  comprehend  the  rule — Notf;.  Feet  multiplied  by  feet 

give  feet — Feet  multiplied   by   inch- 
es give  inches — Feet   multiplied   by 
seconds    give    seconds — Inches    mul- 
EXAMI  LLb.      ^  tiplied  by   inches   give   seconds — In- 

ches    multiplied     by     seconds    give 
1.  Multiply  7  feet,  3  inches,  2      thirds — Seconds     multiplied    by     se- 
seconds,  by  1  foot,  7  inches  and  3      conds  give  fourths, 
seconds. 

OPEKATioN.  Here  I  multiply  the  7f.  Sin.  2"  by 

F.  I.  "  the  If.  in  (he  multiplier,  which  gives 

7  3  2  seconds,  inches. and  feet. 

17  3  Next  I  multiply  the  same  7f.  Sin.  2" 

by   the   7in.   saying   7  times   2  is    14 

'7  3-2'"  which  is  once  12  and  2  over,  which 

4  2  10  2""  (2)  1  set  down  one  place  to  the  right 

19  9  6  hand  that  is  in  the  place  of  thirds  and 

carry  one  to  the  next  place  and  pro- 

Prod.   117  9  116  ceed   in   the   same    manner  with   the 

•iher  terms.    Lastly,   I   multiply    the 

uiultiplicaad  by  the  3"  saying  3  times 

2  are  6,   which  I  set   down  two  places  to  the  right  hand  and  so  proceed 

wifi  the  other  terms  of  the  multiplicand.     The  '^uai   of  all  the  products 

i«  tlif  answer. 


Sect.  III.  8. 

DUODECIMALS. 

2 
F.  I. 

F.  I." 

3 
F.  I. 

F.  I. 

4 

F.  I. 

F.  I. 

3    g  ^  27  9  9  Prod. 

5    8^^^^-^- 

n|^=3. 

189 


5.  Multiply  7f.   lin.  9^  by  7f  6.  Multiply  Of.  Sin.  7"  by  19f. 

Sin.  9"  Sin.   10" 

Prod.   55f.   Sin.   9"  3'"  9""  Prod.    119f.   8'  2"   10'"   10"" 


7.  How  much  wood  in  a  load  which  measures  lOf.  in  lengUi,  3f.  9in. 
in  width,  and  4f.  Sin.  in  height  ;  and  how  much  will  it  cost  at  ^1,33 
per  cord  t      Ans.  1  cord  and  47  solid  /est  over — it  will  cost  j^l  Bids.  3m. 


190  DUODECIMALS.  Sect.  III.  8. 

Or,  we  lYiay  multiply  by  the  feet  as  alread}--  directed,  and  for  the  inches, 
lake  such  parts  of  the  multiplicand,  iic.  as  the  inches  are  aliquot  or  even 
pans  of  a  foot  as  done  in  the  rule  of  Practice. 

a.  How  many  square  feet  in  a  board  of  16  feet  4  inches  in  length,  and 
2  feet  8  inches  wide  ? 

OPERATION.  Here  in  the  first  place  I  multiply 

Ft.     In.  the  16ft.  4in.  by  the  feet  (2)  of  the 

6  indies  i*  i    16      4  multiplier  ;  the   inches   (8)   not  be- 

2      8  ing  an  even  part  of  a  foot,  I  take 

such  as  are  an  even  part ;  thus,  Gin. 

is  half  a  foot,   therefore   divide  the 

multiplicand  by  2  for  6  inches;  and 

that  quotient  by  3  (2in.  is  A  of  Gin.) 

for  2  inches,   all  which  being  added 

Ans.  K>      6  y  give  the  product  of  IG  feet  4  inches, 

multiplied  hy  2  feet  8  inches. 
'.).  Another  board  i>*  18  feet  9  inches  in  length,  and  2  feet  6  inches  wide, 
how  many  square  feet  does  it  contain  ?  Am.  46ft.   lOin.  6" 

By  Practice.  By  Duodecimals. 


32 

'        9 

8 

2 

8  8 

10.  There  is  a  stock  of  16  boards,  12  feet  8  inches  in  length,  and  13 
inches  wide  ;  how  many  feet  of  boards  does  the  stock  contain  ? 

Ans.  205ft.  lOin. 
By  Practice.  By  Duodecimals. 


Sect.  III.  8.         ^t;FPLEMENT  TO  DUODECIMALS.  191 

SUPPLEMENT  TO  DUODECI3IAL8. 


QUESTIONS. 
J.  Of  what  use  are  Duodecimals  ?  To  whom  more  specially  are  they 
useful  ? 

2.  In  what  are  dimensions  taken  ? 

3.  How  do  you  proceed  in  the  multiplication  of  duodecimals  ? 

4.  For  what  number  do  you  carry  ? 

5»  What  do  you  observe  in  regard  to  setting  down  the  product  different 
from  what  is  common  in  the  multiplication  of  other  numbers  ? 

6.  Of  what  term  is  the  product  which  arises  from  the  multiplication  of 

feet  by  inches  ?  feet  by  seconds  ?  inches  by  inches  1  inches  by  sec- 
onds ?  seconds  by  seconds  ? 

7.  In  what  way  can  the  operation  be  varied  ? 

EXERCISES. 
1.    Multiply  76  feet    3  inches  9         2.  What  is  the  product  of  371ft. 
seconds  by  84  feet  7  inches  11  sec-     Sin.  6  seconds,  multiplied  by  181A. 
ends.  lin.  9"  ? 

OPERATION.  Ans.  67242ft.  lOin.  1"  4'"  6"" 

F.  I.     " 
6  inches  is  |)76  3     9 
84  7  11 


76X  4=304  0 

0 

76X  8=608     0 

0 

3X84=  21  0 

0 

9X84=     5  3 

0 

m 

/.  11)           38   1 

10 

6 

"   61)             6  4 

3 

9 

till 

Si)  4- 21)  3  2 

1 

10 

6 

1   7 

0 

11 

3 

1  0 

8 

7 

6 

Prod.  6460  7     18  3 
3.  How  many  square  feet  in  a 
stock  of  12  boards,   17ft.  7'  long, 
and  1ft.  Sin.  wide  ?     Ans.  298ft.  1 1' 


4.  Ho^  many  cubic  feet  of  wood 
in  a  load  6ft.  7'  long,  3ft.  6'  high  and 
3ft.  8'  wide  ?     Ans.  82ft.  5'  (.''  4'" 


192  SUPPLEMENT  TO  DUODECIMALS.       Sect.  IIL  8. 

The  dimensions  of  wainscotting,  paying,   plastering   and  painting  are 
taken  in  feet  and  inches,  and  the  contents  given  in  yards. 
PAINTERS  AND  JOINERS 

To  find  the  dimensions  of  their  work,  take  a  line  and  apply  one  end  of  it 
to  any  corner  of  the  room,  then  measure  the  room,  going  into  every  comer 
with  the  line,  till  you  come  to  the  place  where  you  first  I)egan  ;  then  see 
how  many  feet  and  inches  the  string  contains  ;  this  call  the  Compass  or 
Round,  which  multiplied  into  the  height  of  the  room  and  the  product  di- 
vided by  9,  the  quotient  will  be  the  contents  in  yards. 

EXAMPLES. 

1.  If  the  height  of  a  room  painted  2.  There  is  a  room  wainscotted, 
be  ISft.  4\n.  and  the  compass  84ft.  the  compass  of  which  is  47ft.  3' and 
1  lin.  How  many  square  yards  does  it  the  height  7ft.  6'.  What  is  the  con- 
contain  ?  Am.  liey.  3ft.  3'  8"        tent  in  square  yards  ? 

Ans.  39Y.  3ft.  4'  6" 


GLAZIER'S  WORK  BY  THE  FOOT. 

To  find  the  dijnetisions  of  their  work,  multiply  the  height  of  windows  by 
their  breadth. 

EXAMPLE. 
There  is  a  house  with  4  tiers  of  windows,  and  4  windows  in  a  tier;  the 
height  of  the  first  tier  is  6ft.  8' ;  of  the  second  5ft.  9' ;  of  the  third  4ft.  6' ; 
and  of  the  fourth  3it.  10';  an<l  the  breadth  of  each  is  3ft.  [>' — What  will  the 
glazing- come  to  at  19  cents  per  foot  ?  Ans.  §53,88. 


Sect.  III.  9.  ALLIGATION.  193 

J  9.  ALLIGATION. 


ALLIGATION  is  the  method  of  mixing  two  or  more  simples  of  di/feireAt 
quaiitiee ,  so  tlrat  the  composition  may  b«  of  a  mean  or  middle  quality.  It 
is  of  two  kind3,  Medial  and  Alternate. 

ALLIGATION  MEDIAL, 

AUigation  Medial  is  when  the  quantities  and  prices  of  several  things  3tre 
g-iven  to  find  the  mean  price  of  the  mixture  compounded  of  those  things. 

RULE. 

As  the  sum  of  the  quantities  or  whole  composition  is  to  their  total  value, 
Fo  is  any  part  of  the  composition  to  its  value  or  mean  price. 

EX.iMPLES. 
1.  A  farmer  mingled  19  bushels  of  wheat  at  6s.  per  bushel,  and  40bueh« 
«1s  of  rye  at  4s.  per  bushel,  and  12  bushels  of  barley  at  3s.  per  bushel  te- 
c:e(hcr.     I  demand  what  a  bushel  of  this  mixture  is  worth  ? 

OPERATION. 

Bush.  s.     £,.     s.  Buih.     £.     s.     Bush. 

19  Wieat  at,     .  6  is  5     14  As  71    :   15     10  :  :   I 

40  Rye         —  4  —  8  20 

12  Barley    —  3  —  1      16                         

sum  of  the—                           71)310(45.  4d.  l^{q.  Ajis 

simples    71    Total  value.  15     10  284 

?.  A  Refiner  having  5/6.  of  silver  bullion,  26 

o(  V.oz.  fine,  lOlb.  of  loz.  fine,   and  I5lb.  of  12 

Goz.  fine,  would  melt  all  together  ;  I  demand  

what  finaness  lib.  of  this  mass  shall  be  ?  )312(4i. 

Ans.  6o2.    ISp-X'ts.   Sgrs.  Jine.  284 

28 
4 

)112(ly. 
71 

41 


A  \ 


194  ALLIGATION.  Sect.  IIL  9, 


ALLIGATION  ALTERNATE, 

Is  the  method  of  finding  what  quantity  of  any  number  of  simples,  whose 
rates  are  given  will  compose  a  mixture  of  a  given  rate,  it  is  therefore,  the 
reverse  of  Alligation  Medial,  and  may  be  proved  by  it. 

RULE. 

1 .  Write  the  prices  of  the  simples,  the  least  uppermost,  &c.  ia  a  column 
under  each  other. 

2.  Connect  with  a  continued  line  the  price  of  each  simple  or  ingredient, 
which  is  less  than  that  of  the  compound,  with  one  or  any  number  of  those 
that  are  greater  than  the  compound,  and  each  greater  rate  or  price  with' 
one  or  any  number  of  those  that  are  less. 

3.  Write  the  difference  between  the  mean  rate  or  price  and  that  of  each 
of  the  simples  opposite  to  the  rates  with  which  they  are  connected. 

4.  Then  if  only  one  difference  stand  against  any  rate  it  will  be  the 
quantity  belonging  to  that  rate,  but  if  there  be  several,  their  sum  will  be 
the  quantity. 

Note.  Questions  in  this  rule  admit  of  as  many  various  answers  as  there 
are  various  ways  of  connecting  the  rates  of  the  ingredients  together. 

EXAMPLES. 
A  goldsmith  would  mix  gold  of  18  carats  fine  with  some  of  16,  19,  22, 
and  24  carats  fine,  so  that  the  compound  may  be  20  carats  fine  ;  what 
quantity  of  each  must  he  take  ? 

OPERATION.  PROOF. 

oz.  car.  fine. 

4  of  gold  at  16^  16X4=64 

18  18X2=36 


Mix  ^0  car.  {  19- 

22- 


f  16 X  4 


2 
2X1 


(^24 '  4 


2 19  \Jins.  19X2=38 

3  .....  22  22X3=66 

4 24  J  24X4=96 


02 


15  ^Q  carats  fine.     15)300(20  car. /ne. 

2.  A  druggist  had  several  sorts  of  tea,  viz.  one  sort  at  125.  per  lb.  another 
sort  at  1 1*.  a  third  at  9s.  and  a  fourth  at  8s.  per  lb.  I  demand  how  much 
of  each  sort  he  must  mix  together,  that  the  whole  quantity  may  be  afforded 
at  105.  per  lb. 

Ih.     s.p.lb. 
'3  at  12 

I    Aiis.^C  1'  'A         2    Ajis.{^  ^l  ^g         3    Ans. 

8 
s.p.lb. 
at  12 

4    Ans.  <f  o  - 1   ^  A         5    Ans.  ^  o  \     a         6    •^«*- 

at     8 
7.  .fins.  3lb.  of  each  sort. 

Note.  These  serpen  answers  arise  from  as  many  different  ways  of  linking 
the  rates  of  the  simples  tog'ether. 


lb 

o 

at 

s.p.lb, 
12 

1 

at 

11 

1 

at 

9 

2 

at 

8 

lb 

1 

at 

s.p.lb. 
12 

3 

at 

11 

3 

at 

9 

1 

at 

8 

Ih. 
'1 

5 

at 

.p.lb. 
12 

12 

at 

11 

|2 

at 

9 

'l 

at 

8 

lb. 

'2 

s 
at 

.p.lb. 
12 

3 

at 

11 

M 

at 

9 

3 

at 

8 

Sect.  III.  9i 


ALLIGATION. 


195 


Cass  2. 
fVhen  the  rates  of  all  the  ingredients,  the  quantify  of  but  one  of  them,  and 
the  mean  rate  of  tkc  znhole  mixture  are  given  to  find  the  several  quantities  of 
the  rest,  in  proportion  to  the  given  quantity;  take  the  difference  betweeo 
each  price  and  the  mean  rate  aa  before.     Then  say, 

As  the  difference  of  that  simple  whose  quantity  is  given, 

Is  to  the  given  quantity, 

So  is  the  rest  of  the  differences  severally  ; 

To  the  several  quantities  required. 

EXAMPLES. 
i.  How  much  wine,  at  80  cents,  at  88,  and  92  cents  per  gallon  must  be 
mixed  with  four  gallons  of  wine  at  75  cents  per  gallon,  so  that  the  mixture 
may  be  worth  86  cents  per  gallon? 

OPERATION. 

6+  2=  8  stands  against  the  given  quantity. 

2+  6=  8 

6+11  =  17 

ll-f  0=17 

gal. 


As  8 


cts. 

at  80 


per  gal.     Tlie  ansxcer. 


Ji  — 92 

2.  A  man  being  determined  to  mix  10  bushels  of  wheat  at  4s.  per  bushel, 
with  rye  at  3«.  with  barley  at  2s.  and  with  oats  at  Is.  per  bushel  ;  I  demand 
how  much  rye,  barley  and  oats  must  be  mixed  with  the  10  bushels  of  wheat 
that  the  whole  may  be  sold  at  28d.  per  bushel  ? 


1    Ans. 


4    Ans. 


B.p. 
2  2  of  Rye 
5  0  -  Barley 
12  2  -  Oats 

B. 

10  of  Rye 
10  -  Barley 
15  -  Oats 


2   Ans. 


5    Ans. 


7.  Ans. 


B.  B. 

40  of  Rye  (   8  of  Rye 

50  -  Barley  3   Ans.  ?  10  -   Barley 

20  -   Oats  (  14  -   Oats 

B.p.  B. 

12  2  of  Rye  i    2  of  Rye 

5  0  -  Barley  6  Ans.  lu-  Barley 

17  2  -  Oats  {  10  -  Oats 

B. 

50  of  Rye  - , 
70  -   Barley 
20  -  Oats 


;»96  ALLIGATION.-  SfccT.  IIL  a 

Casc  3. 
R7/cn   the  rates  of  the   .several  tngrcdientSy  (he  quantity  to  be  compounded, 
and  the  mean  rate  of  the  n-liole  inixtvre  are  giuen,   to  find  ho7i.'  much  of  each 
^"Tt  n-ill  make  up  the  quantity,  find  the  differences  between  the  mean  rate, 
!^c.  as  in  Case  1.     Then, 

^\s  the  sum  of  the  fjiiantitics  or  difTorenoo!', 
Is  to  the  given  quantity  or  whole  coniposition  ; 
Sais  the  ditlerencc  of  each  rate. 
To  the  required  qtianlity  of  each  rate. 

EXAMPLES. 

1.  How  many  gallons  of  crater,  of  no  value,  must  be  nri:xed  with  brandy, 
at  one  dollar  twenty  cents  j)er  ^nllon,  po  as  to  fill  a  vessel  of  75  gallons, 
that  may  be  afforded  ut  92  cerrts  per  gallon  ? 

OrERATJO>f. 

Gal.  Gal.       Gal. 

r^^  C       0->28  Gnl.     Gal.        <■  20  :    11 -\  of  Hater. 

"''  \  1,20-^92  As    120  i  75  :  :     }  92   :  57^  -    Brarldy. 

Sum  1 30  75  given  quantity. 

2.  Suppose  I  have  4  sorts  of  currants  of  Bd.  12c?.  IBd.  and  22f/.  per  lb. 
of  wliich  I  would  mix  120lh.  and  ?n  much  of  each  sort  as  to  sell  them  at  16^. 
jrer  lb.  how  much  of  each  must  I  take  ? 

lb.    at    d. 


'^"*'  <  Qd  1  fi  /  P^^  '^* 


".  A  Grocer  ha<»4urrant3  of  4d.  Gd.  9d.  and  ild.  per  lb.  and" he  wouM 
make  a  mixture  of  2401b.  so  that  it  might  be  afforded  at  8d.  per  lb.  how 
much  of  each  sort  roust  he  take  ? 


lb. 

at    d 

72 

—     4 

24 

—    e 

48 

f) 

PG 

—  n 

./Ins.  «( ■;  n/  per  lb. 


SECT.  III.  9.         SUPPLEMENT  TO  ALWGATION.  197 

SUPPLEMENT  TO  ALLIGATION. 


QUESTIONS. 

1.  What  is  Alligation? 

2.  Of  Iiow  many  kinds  is  Alligatioa  ? 

3.  Wliat  is  Alligation  Medial  ? 

4.  What  is  the  rule  for  operating  ? 

5.  What  is  Alligation  ALTl;p.^ATC  ? 

6.  When  a  number  of  ingredienls  of  different  priced  are  mixed  together. 

How  do  we  proceed  to  find  the  mean  price  of  the  compound  or 
mixture  ? 
•7.  When  one  of  the  ingredients  is  limited  to  a  certain  quantity,  v;hat  is 
the  nielhod  of  procedure  ? 

8.  When  the  whole  composition  is  limited  to  a  certain  q<ianti<y,  how  ffo 

you  proceed  ? 

9.  How  is  Alligation  proved  ? 

EXERCISES. 
I.  A  Grocer  would  mix  three  2.  A  Goldsmith  has  several  sorts 
sorts  of  sugar  together;  one  sort  of  gold  ;  some  of  24  carats  fine,  some, 
at  10c?.  per  lb.  another  at  Id.  and  of  22  and  some  of  18  carats  fine,  and 
another  at  6d.  how  much  of  each  he  would  have  compounded  of  these 
sort  must  he  take  that  the  mixture  sorts  the  quantity  of  60o-:.  of  20  Ca- 
may be  sbld  for  8c?.  per  lb.  ?  rats  fine  ;  I  demand  hpw  much  of 
Ans.  3/6.  at  lOJ.  2/6.  at  Id.     each  sort  must  he  have  ? 

4*  2/6.  at  Qd.  Ans.  12or.  SI  carats  fine,  12  c/22 

carats  fine,  and  36  at  Id  car.fijie. 


1«8 


POSITION. 
#  10.  POSITION. 


Sect.  III.  10 


POSITION  is  a  rule  which  by  false  or  supposed  numbers,  taken  at 
pleasure,  discovers  the  true  one  required.  It  is  of  two  kinds,  Single  and 
Double. 

SINGLE  POSITION. 

Is  the  working  of  one  supposed  number,  as  if  it  were  the  true  one,  to 
find  the  true  number. 

RULE. 

1.  Take  any  number  and  perform  the  same  operations  with  it  as  are  de- 
scribed to  be  performed  in  the  question. 

2.  Then  say  as  the  sum  of  the  errors  is  to  the  given  sum,  so  is  the  sup- 
posed number  to  the  true  one  required. 

Proof.  Add  the  several  parts  of  the  sum  together,  and  if  it  agree  with 
the  sum,  it  is  right. 

EXAMPLES. 
1 .  Two  men,  A  and  B,  having  found  a  bag  of  money,  disputed  who  should 
have  it.    A  said  the  half,  third,  and  one  fourth  of  the  money  made  130  dol- 
lars, and  if  B  could  tell  how  much  was  in  it,  he  should  have  it  all,  other- 
wise he  should  have  nothing  ;  I  demand  how  much  was  in  the  bag  ? 

OPERATION. 

Suppose  60  dollars.  As  65  :   130  :  :  60 

60 

The  half    SO  

—    third  20  66)7800(120  dolls,  the  aitD^'er. 

-^   fourth  15  65 


66 


130 
130 

000 


2  A  B  and  C  talking  of  their  ages, 
B  said  his  age  was  once  and  a  half 
the  age  of  A  ;  and  C  said  his  age 
was  twice  and  one  tenth  the  age  of 
both,  and  that  the  sum  of  their  ages 
was  93  ;  what  was  the  age  of  each  ? 
A's  12,  B's  18,  C's  63  years. 


3.  A  person  having  spent  \  and  ^ 
of  his  money,  had  £26|  left,  what 
bad  he  at  first?  Ans.  £,1^0. 


4.  Seven  eighth*  of  a  certain  num- 
ber exceeds  four  fifths  by  6  ;  whal 
is  that  number  ?  Jins.  80. 


Sect.  III.  10.  DOUBLE  POSITION.  199 


DOUBLE  POSITION. 

DOUBLE  POSITION  is  that  which  discovers  the  true  number,  or  num- 
ber sought,  by  making  use  of  two  supposed  numbers, 

RULE. 

1.  Take  any  two  numbers  and  proceed  with  them  according  to  the  con- 
ditions of  the  question. 

il.  Place  each  error  against  its  respective  position  or  supposed  Rumber  ; 
if  the  error  be  too  great,  mark  it  with  +  ;  if  too  small,  with  — 

3.  Multiply  them  cross-wise,  the  first  position  by  the  last  error,  and  the 
last  position  by  the  first  error. 

4.  If  they  be  alike,  that  is,  both  greater,  or  both  less  than  the  given 
number,  divide  the  difference  of  the  products  by  the  difference  of  the 
errors,  and  the  quotient  will  be  the  answer;  but  if  the  errors  be  unlike, 
divide  the  sum  of  the  products  by  the  sum  of  the  errors,  and  the  quotient 
will  be  the  answer. 

EXAMPLES. 

U  A  man  lying  at  the  point  of  death,  left  to  his  three  sons  all  his  estate, 
♦Hz.  to  F  half  wanting  50  dollars  ;  to  G  one  third  ;  and  to  H  the  rest,  which 
was  10  dollars  less  than  the  share  of  G.  I  demand  the  sum  left,  and  each 
Bon's  share. 

OPEJIATION. 

Suppose  the  sum  300  dollars.       Again  suppose  the  sum  900  dollars. 


Then  300-J-2— 50=100  F's  part.  Then  900-7-2—50=400  F's  part. 

300-f-  3=100  G's  part.  900-H  3=300  G's  part. 

G's  part    100—10=  90  H's  part.  G'a  part   300—10=290  H's  part. 

Sum  of  all  their  parts  290  Sum  of  all  their  parts  990 


X 


Error  10 —  Error  90-{- 

Suppose.     Errors.  Having  proceeded  with  the  sup- 

300  10 —  posed  numbers    according    to   the 

conditions  of  the  question,  the  siim 
of  all  their  parts  must  be  subtracted 
900  90-|-  from  the   supposed  number ;  thus 

the  290  is  subtracted  from  300,  the 

9000     27000  supposed  number,  kc. 
27000 
Dollars. 

^vHZvI.}    100)36000(360  Ans. 
The  divisor  is  the  sum  of  the  errors  90+and  10 — 

2.  There  is  a  fich  whose  head  is  10  feet  long;  his  tail  as  long  a?  bi5 
head  and  half  the  len£,-th  of  his  body,  and  his  body  as  long  as  his  fu-<d 
and  tail ;  what  is  the  whole  length  of  the  fish  ?  Ans.  8U  fexil. 


fOO  DOUBLE  POSITION.  Sect.  111.   10. 

'J.  A  certain  man  having  driven  his  swine  to  n»arkct,  via.  hogs,  sows  and 
pigs,  received  for  them  all  £50  being  paid  lor  every  hog  lla.  for  every 
60W  165.  for  every  pig  2s.  ;  there  were  as  many  hogs  as  sows,  and  for 
every  sow  there  were  three  pigs  ;  I  demand  ho\v  many  there  were  of 
each  sort  ?  »3"s-  2o  hogHy  25$oxs,  and  75/><j^s. 


4.  A  and  B  laid  out  equal  sums  of  money  in  trade  ;  A  gained  a  sum 
equal  to  |  of  his  stock,  and  B  lost  225  dollars  i  then  A's  naoney  was  double 
that  of  B's ;  what  did  each  one  lay  out?  Ms.  ^600. 


5.  A  and  B  have  the  same  income  ;  A  saves  i  of  his  ;  but  B  by  spend- 
ing 30  dollars  per  annum  more  than  A,  at  the  end  of  8  years  finds  himself 
40  dollars  iu  debt ;  what  is  their  income,  arid  what  does  each  spend  per 
annum  ?  '^I 

Ans.  Thdr  income  is  ^200 per  am-  A  spends  $1!^,  «$'  B  205  per  ann. 


SetT.  IIJ.  lU  DISCOUNT.  2U1 

i  U.  DISCOUNT. 


DISCOUNT  is  an  aHoivance  made  for  the  payment  of  any  sum  of 
(iiunev  before  it  becomes  due,  and  io  the  difference  bet-iveen  that  sum,  due 
sometime  hence,  and  its  preseiit  north. 

The  presmt  wo;'/i  oi"  I'.iiy  sum  ur  debt  due  some  time  hence,  is  such  a 
snia  as,  it'  put  to  interes',  would  in  tiiat  time  and  at  tlie  rate  per  cent  lor 
which  the  discouut  is  to  be  made,  amount  to  tlie  sum  or  debt  tlicn  duo. 

RULE. 
As  tlie   amoant  ol    loO  dollars  t"i>r  the  given  time  and  rat-e  is  to  100  dol- 
lars, so  is  the  given  sum  to  its  present  worth,  which  subtracted  from  the 
given  sum  leaves  the  discount. 

EXAMPLES. 
1.   What  is  the  discount  of  1^321,63         2.  What  is  the  present  worth  of 
due  4  years  hence  at  0  per  cent?  ^42G  payable    in   4  years  and  IJ 

(Jays,  discounting-  at  the  rate  of  .*} 
o^•l:;RATIo^'.  percent?  .i/is.  ^354,51^. 

Dolls. 
6  interest  of  100  doUa. 
4  years.  [1  year. 

24 
100 

124  amount. 


Then,  as  124  :    100  :  :  321,63 
321,63 


124)32163,00(259,379 

321,03  given  sum. 
259,379  present  worth. 


Mns.  G2,25I  discount. 

i  r2.  EaUATION  OF  PAYMENTS. 


EQUATION  OF  PAYMENTS  is  the  fmding  of  a  time  to  pay  at  once, 
a.cveral  debts  due  at  different  times,  su  that  neither  party  shall  sustain  loss.> 

RCLE. 
Sfulfiply  each  payment  by  the  time  at  which  it  is  due  ;  then  divide  the 
?um  of  the  pnxhicLs  by  the  ^■um  t.f  the  payments,  and  the  quolient  will  be 
the  c<ii;uted  timt;. 

H  h 


t02  EQUATION  OF  PAYMENTS.  Ssct.  III.  li. 

EXAMPLES. 
1.    A  owes  B  136  dollars  to  be         2.   I  owe  ^65,125,  to  be  paid  i  in 

paid  in  10  months  ;   96  dollars  to  be  3   months,  ^  in  5  months,  \  in    10 

paid  in  7  months;   and  260   to  be  months,    and   the    remainder    in    14 

paid  in  4  months ;  what  is  the  equa-  months,    at    what   time    ought    the 

ted  time  for  the    payment   of  the  whole  to  be  paid  ? 
whole  ?  -fins.  6}  months. 

OPERATION. 

136x  10=1360 

96x  7=  672 

SGux  4=1040 


402  3072 

492)3072(6  months. 
2952 


120 
30 


492)3600(7  days. 
3444 

166 


3.  A  merchant  has  owing  to  him         4.  A  merchant  owes  me  900  dol- 

£300,  to  be   paid  as  follows ;   £50  lars,  to  be  paid  in  96  days,  130  dol- 

at  2  months,  £100  at  5  months,  and  lara  in    120  days,  500  dollars  in  80 

the    rest  at  8  months  ;   and  it  is  a-  days,  1267  dollars  in  27  days;  what 

greed  to  make  one  payment  of  the  is  the  mean  time  for  the  payment  of 

whole  ;    I  demand  when   that   time  the  whole  ? 
must  be  ?                  Ans.  6  months.  Ans.  63  days,  very  nearly. 


Sect.  III.  13. 


GUAGING. 


SOS 


^  13.  GUAGING. 


GUAGING  is  talung  the  dimensions  of  a  cask  in  inches  to  find  its  cont 
tents  in  gallons  by  the  following 

METHOD. 

1.  Add  two  thirds  of  the  difference  between  the  head  and  bung  diam- 
eters to  the  head  diameter  for  the  mean  diameter ;  but  if  the  staves  be 
but  little  curving  from  the  bead  to  the  bung,  add  only  six  tenths  of  this 
difference. 

2,  Square  the  mean  diameter,  which  multiplied  by  the  length  of  the  cask, 
and  the  product  divided  by  294,  for  wine,  or  by  369  for  ale,  the  quotient 
will  be  the  answer  in  gallons. 

EXAMPLES. 

«.  How  many  ale  or  beer  gallons  will  a  cask  hold,  whose  bung  diameter 
1  inches,  head  diameter  25  inches,  and  whose  length  is  36  inches  ? 

OPERATION. 

31  Bung  diam.  25  head  diam. 

25  Head  diam.     4  Two  thirds  difference. 


S  Difference.    29  Mean  diam. 
29 

261 
58 


841  Square  of  m»an  diam. 
36  Length. 

6046 
2523 


36&)30276(04  galls.  HH  ?'*• 


Note  1.  In  taking  the 
length  of  the  casks,  an  al- 
lowance must  be  made  for 
the  thickness  for  both 
heads  of  one  inch,  li inch- 
es, or  2  inches  according 
to  the  size  of  the  cask. 

Note  2.  The  head  di- 
ameter must  be  taken  close 
to  the  chimes,  and  for 
small  casks,  add  3  tenths 
of  an  inch  ;  for  casks  of  40 
or  60  gallons,  4  tenths, 
and  for  larger  casks,  5  or  6 
tenths,  and  the  sum  will  be 
very  nearly  the  head  di- 
ameter within. 


$  14.  MECHANICAL  POWERS. 


1.  OF  THE  LEVER. 

To  find  -achat  weight  may  be  raised  or  balanced  by  any  given  porver,  Say  as 
the  distance  between  the  body  to  be  raised  or  balanced,  and  the  fidcrum 
or  prop,  is  to  the  distance  between  the  prop  and  the  point  where  the  power 
is  applied  ;  so  is  the  power  to  the  weight  which  it  will  balance  or  ruine: 


SOf  '  MECHANICAL  POWERS.  Sect.  III.  U. 

EXAMPLE. 
IF  a  man  weigbing  150lb.  rest  on  the  end  of  a  lever  12  feet  long,  what 
weight  will  he  balance  on  the  other  end,  supposing  the  prop.J-i-  feet  from 
the  weight  ? 

1 2  feet  the  L&vcr. 

1,5  distance  of  the  weight  from  the  fulcrum. 

10,5  distance  from  ibe  fulcrum  to  the  man.     Therefore, 
Feet.     Feet.         Lb.       lb. 
As   1,5  :    10,5  :  :  150  :  1050  Am. 

2.  OF  THE  WHEEL  AND  AXLE. 

As  tlie  diatneler  of  the  axle  is  to  the  diameter  of  tlie  wheel,  jso  is  th.e 
po'.ve  r  i'pjdiod  to  ihe  wheel,  to  the  weight  suspended  bj  the  axle. 

EXAMPLES.  ^ 

1.  A  mechauic  wishes  to  make  a  windlass  in  such  a  manner,  as  that  m^ 
ajrplied  to  the  wheel  should  be  equal  to  12  suspended  on  the  axle  ;  now 
suppdsing  the  axle  4  inches  diameter,  required  the  diamcfcr  of  the  wheel  ? 

lb.       in.       lb.       in. 
As   1     :     4  :  :    12    ;    48    Ans.  or  diameter  of  the  wheel. 

2.  Suppose  the  diameter  of  the  axle  C  inches,  and  that  of  the  wheel  60 
inches,  what  power  at  the  wheel  will  balance  101b.  at  the  axle  '.     Ans.  lib. 

3.  OF  THE  SCREIK 

The  power  is  to  (he  \veight  to  be  raised  as  the  distance  be^tween  2 
threads  of  the  screw  is  to  the  circuniferencc  of  a  circle  described  by  tlie 
po.rer  applied  at  the  end  of  the  lever. 

Norr,  1.  To  find  the  circumference  of  the  circle  drscribed  by  the  end  of 
the  lever,  multiply  the  double  of  the  lever  by  3,14159,  the  product  will  bo 
the  circ\imference. 

Nor£  2.  It  is  usual  to  abate  ^  of  the  effect  of  the  macliine  for  friction. 

EXAMPLES. 
There  is  a  screw  whose  threads  are  an  inch  asunder  ;  the  lever  by  which 
it  is  turned,  is  36  inches  long,  and  the  weight  to  be  i-aised  a  ton,  or  22401b. 
What  power  or  force  ui'jst  be  applied  to  the  end  of  the  lever,  sufficient  to 
turn  the  screw  that  is  to  rajse  the  weiglit? 

Tht  lever  3Gx2=72<3,14J59=226,194+iAc  circumference. 
cii'curaf.         in.         lb.         lb. 
TAew,  03  220,194     :     1   :  :  2240  :  0,903 

PROBLEMS. 

1.  The  illnvieter  of  a  circle  being  give7i  to  find  the  circwnfereyice,  multiply 
the  diameter  by  3,1415'.^  ;  the  product  will  be  the  circumference. 

2.  To  fnd  ihe  area  of  a  circle,  the  diameter  being  given,  multiply  the 
Sfpjijire  of  the  diameter  by  ,78539^  ;  the  product  is  the  area. 

3.  To  mtaattrc  the  soliditij  of  any  irregular  body,  Xi'hose  dimensions  cannot 
be  taken,  put  the  body  into  some  regular  vessel  and  till  it  with  water,  th^u 
taking  out  the  body,  nj'/asure  the  frill  of  water  in  the  vessel ;  if  the  vessel 
be  square,  multiply  the  side  by  itself,  and  the  product  by  the  fall  of  water, 
ivhich  gives  the  solid  contents  of  the  irregular  body. 


MISC£LLA.\EOUS  at'ESTIOINS. 


1.  THE  Northern  Lights  were  first  observed  in  London  1560  ;  bow  ma- 
ny years  since  ? 

J^.  V/ hat  number  mulliplisd  by  43  produces  88150  ?  Jlns.  2050. 

BP^.  If  a  cannon  may  be  discharg'ed  twice  with  61b.  of  powder,  how  many 
times  will  7C.  3<jrs.  lllhs.  discharge  the  same  piece  ?       Ans.  295  times. 

4.  Reduce  14  guineas  and  £75  13.».  6^d.  to  Federal  Jlonev. 

Ans.'$^n,593. 

5.  What  is  the  interest  of  j^70,47   one  year  and  fire  months  ? 

Ans.  ^6,754. 

G.  A  owed  B  <J3I7,19,  for  which  he  ffave  his  note  on  interest,  bearing 
date  July  12th,  1797.  On  the  back  of  the  note  arc  these  several  endorse- 
ments, vir. 

Oct.  17th,  1797,  Received  in  cash  ^61.10. 

March  20th,  1790,  Received  \lCwt.  l)cef,  at  ^4,33  per  cwt. 

Jan,  1st.  IJ-JOO,  Received  in  cash,  $84, 

What  was  there  due  tVom  A  to  B  of  principal  and  interest,  Sept.  JG<h, 
1801  ?  Ans.  ^144.303. 

7.  What  cost  13A  yards  ofllannel  at  if.  S^d.  per  yard  ?  Ans.' £\  3».  Ofr?. 

8.  What  must  I  give  for  3C'a'.  2<jrs.  13/6.  of  cheese  at  7  ctg.  per  lb,  ? 

Ans.  §28,33. 

9.  What  will  35  yards  of  broadcloth  cost  at  23s.  Cd.  per  yard  ? 

Av's.  £41  2j.  6a. 

10.  What  will  be  the  cct  of  a  loin  of  venl,  weighing  IG^lb.  at  2JrZ.  per  lb.  ? 

Arts'  3s.  53d. 

1 1 .  What  will  C7i;^.  of  Utllow  cost  at  9|(^  per  lb.  ?     Ans.  £3  75,  3d. 
]'J.   What  will  19G  yards  of  tape  cost  at  3  farthings  per  vard  ? 

■  Ans.  \2s.3d. 

13.  What  will  56  busheU  of  oats  cost  at  2s.  3^d.  per  bt7?hel  ? 

.^.ns.  £6  Ss.  4rf. 

14.  At  £3  7.'.  f>d.  per  cwt.  for  •:ti2:ir,  what  is  that  per  lb.  ?      Ans.  Id. 

15.  How  much  m  length  of  a  board  that  is  10  inches  wide  will  it  require 
to  make  a  square  foot  ?  Ans.  1 4 j\  feet. 

16.  How  many  square  feet  in  aboard  1  foot  3  inches  wide,  and  14  feet  9 
jrKhp=  fong  ?  .Ins.  IS/,  o  3" 

17.  How  much  wood  in  a  load  9  feet  long,  31  wide,  and  2  feet  9  inches 
Irigh  ?  Ans.  86/.  7'  6" 

13.  At  ^1.33  per  yard  for  clotb,  what  nousti  give  for  72  A'ards  ? 

Ans.  ^95,76. 


SOS  MISCELLANEOUS  QUESTIONS. 

19.  If  2i  cwt.  of  cotton  wool  cost  £11  17s.  6d.  what  is  that  per  lb  ? 

^ns.   llirf. 

20.  If  1832^  gallons  of  wine  cost  £44  6s,  what  is  that  per  gallon  ?  Aiis.  3|i 

21.  What  will  o3ilh.  of  beef  cost  at  5cts.  5m.  per  lb.  ?     Ans.  ^2,942. 

22.  What  will  50  bushels  of  potatoes  cost  at  21  cents  per  bushel  ? 

Ans.  ^10,50. 

23.  At  $10,76  per  cwt.  for  sugar,  what  is  that  per  lb.  ?   Aiu.  ^0,096. 

24.  What  will  be  a  man's  wages  for  6  months,  at  43  cents  per  day, 
working  5A  days  per  week  ?  Ans.^6\,49. 

25.  What  must  1  give  for  pasturing  my  horse  19  weeks,  at  33  cents  per 
week  ?  Ans.  $6,27. 

26.  Mow  many  revolutions  doe?  the  moon  perform  in  144  years,  2  days, 
10  hours.     One  revolution  being  in  27  days,  7h.  43m.  Ans.  1925. 

27.  What  will  7  pieces  of  cloth  containing  27  yards  each,  come  to  at 
15s.  i^ii.  per  yard  ?  Ans.  £145  ds.  lOirf. 

28.  A  man  spends  23  dolls.  69  cents,  6  mills,  in  a  year,  what  is  that  per 
day  ?      .  Ans.  i?0,064j^ 

29.  Suppose  the  Legislature  of  this  State  sjould  grant  a  tax  of  7  cent^^ 
mills  on  a  dollar,  nliat  will  a  man's  tax!  be,  who  is  142  dollars  40  cents  on 
the  list  ?  .^ny.  S  10,395. 

30.  A  Bankrupt,  uhose  effects  are  3948  dollars,  can  pay  his  creditors 
but  2G  Gts.  5  mills  on  the  dollar.  What  does  he  owe  ?    Ayis.  g  13852,63 1. 

31.  .Suppose  a  cistern  having  a  pipe  that  conveys  4  gallons,  2qts.  into  il 
in  an  hour,  has  another  that  lets  out  2  gallons,  Iqt.  Ipt.  in  an  hour,  if  the 
cistern  contains  84  gallons,  iu  what  time  will  it  be  filled  ? 

Ans.  S9h.  31m.  45\^s. 

32.  If  80  dollars  worth  of  provisions  will  serve  20  men  25  days,  what 
number  of  men  will  the  same  provisions  serve  10  days  ?       Ans.  50  men. 

33.  If  6  men  spend  16  dollars  7  cents  in  40  days,  how  long  will  135  men 
be  spending  100  dollars  ?  Ans.  11  days,  Ih.  30m.  10s. 

34.  A  bridge  built  across  a  river  in  6  months,  by  45  men,  was  washed 
away  by  the  current ;  required  the  number  of  workmen  sufficient  to  build 
another  of  twice  as  much  worth  in  4  months  ?  Ans.  135  men. 

35.  Four  men.  A,  B,  C,  &.  D,  found  a  purse  of  money  containing  12  dol- 
lars, they  agree  that  A  shall  have  one  third,  B  one  fourth,  C  one  sixth  and 
D  one  eighth  of  it,  what  must  each  man  have  according  to  this  agreement? 

Ans.  A's  share  $4,511^.     B's  share,  |3,4284. 
C's  share  $2,285f     D's  share,  |l,714f. 

36.  A  certain  usurer  lent  £90  for  12  months,  and  received  principal  and 
interest  £95  8s.     I  demand  at  what  rate  per  cent  he  received  interest. 

Ans.  6  per  cent. 

37.  If  a  gentleman  hare  an  estate  of  £1000  per  annum,  how  much  may 
he  spend  per  day  to  lay  up  three  score  guineas  at  the  year's  end  ? 

A71S.  £2  105.  2d.  IHq. 

38.  What  is  the  length  of  a  road,  which  being  33  feet  wide  contains  an 
acre  ?  Ans.  80  rods  in  length. 

39.  Required  a  number  from  which  if  7  be  subtracted,  and  the  remain- 
der be  divided  by  8,  and  the  quotient  be  multiplied  by  5,  and  4  added  to 
th.""  T>ro.'iiict,  the  square  root  of  the  sum  extracted,  and  three  fourths  of  that 
root  cubed,  the  cube  divided  by  9,  the  last  quotient  may  be  24  ?  Ans.  103. 

40.  If  a  quarter  of  wheat  aft'orJs  GO  ten  jxnny  loaves,  how  many  eight 
peunj'  loaves  may  be  obtaijiod  from  li'i  Jins.  75  ei^ht penny  loames.    . 


MISCELLANEOUS  QUESTIONS.  207 

41.  If  the  carriage  of  7  cwt.  2qr.  for  105  miles  he  £l  5s.  how  far  loAy  5 
cwt.  Iqr.  be  carried  I'or  the  s<iine  money  ?  Arts.  150  miles. 

42.  If  50  men  consume  15  buc-hels  of  grain  in  40  days,  how  much  will 
30  men  consume  in  sixty  cla3'3 '?  ;  Ans.  ]3^  bushels. 

43.  On  the  same  supposition,  how  long  will  50  bushels  maintain  64  men  ? 
:  .dns.  104  days,  4  hours. 

'44.  A  gentleman  having  50s.  to  pay  among  his  laborers  lor  a  day's  work, 
would  give  to  every  boy  Kd.  to  every  woman  8d.  and  to  every  man  ICd. 
the  number  of  boys,  women  and  men  was  the  same  ;  I  demand  the  number 
of  each  ?  .  Aus.  20. 

45.  A  j;^cntlcman  had  £,7  17s.  6(7.  to  pay  among  his  laborers;  to  every 
boy  he  <.!:ave  6d.  to  every  woman  Sd.  and  to  every  man  16(/.  and  there  were 
for  every  boy  tlsree  women,  and  for  every  woman  2  men  ;  1  demand  the 
miDiber  of  each  ?  Ans.  15  boys,  45  wome?},  and  90  men. 

46.  'I'bree  Gardeners,  A,  B,  and  C,  having  bought  a  piece  of  ground, 
find  the  profits  of  it  amount  to  120^0  per  annum.  Now  the  sum  of  money 
vv^ich  they  laid  down  wa<  in  such  proportion,  that  as  often  as  A  paid  bji  B 
p*d  l£,  An^  as  often  as  B  paid  4^  ^  paid  6jC-  I  demand  how  much  each  niaii 
niUHthave  per  annum  of  the  gain  1  Ans.  A  £'-6  13s.  Ad.  B£37  6s.  8(/.  CiJ56. 

47.  A  young  man  received  210i;^  which  was  2-3cis.of  his  eldest  brother's 
portion  :  now  tliree  tinios  the  eldest  brother's  portion  was  half  the  father's 
estate  ;  I  demand  how  much  the  estate  was  ?  Ans.  £1890. 

40.  Two  men  dei)art  both  from  one  place,  the  one  goes  North,  and  the 
other  South  ;  the  one  i;-oes  7  miles  a  day,  the  other  11  miles  a  day  ;  how  far 
are  they  distant  the  12th  day  after  their  departure  ?         Ans.  216  miles. 

49.  Tuo  men  depart  both  from  one  place,  and  both  ^o  the  same  road  ; 
the  one  travels  12  miles  every  day,  the  other  17  miles  every  day  ;  how  far 
are  tiiey  distant  the  10th  day  after  their  departure  ?  Ans.  50  miles. 

50.  The  river  Fo  is  1000  feet  broad,  and  10  feet  deep,  and  it  runs  at  the 
rate  of  4  miles  an  hour.  In  what  time  will  it  discharge  a  cubic  mile  of  wa- 
ter (reckoning  5000  feet  fo  the  mile)  into  the  sea  ?  Ans.  26  days,  1  hour. 

51.  If  the  country  which  supplies  the  river  Po  with  water,  be  380  miles 
long  and  120  broad,  and  the  whole  land  upon  the  surface  of  the  earth  be 
62,700,000  square  miles,  and  if  the  quantity  of  water  discharged  by  the  riv- 
ers into  the  sea  be  every  where  proportional  to  the  extent  of  land  by  which 
the  rivers  are  sup])lied  ;  how  many  times  greater  than  the  Po  will  the  whole 
amount  of  the  river?  be  ?  Jhis.  1375  times. 

52.  Upon  tlie  same  sujipo^ition,  what  quantity  of  water  altogether  will  be 
discharged  by  all  the  rivers  into  the  sea  in  a  year  ]   Ans.  19272  cuhic  miles. 

53.  If  the  proportion  of  the  sea  on  the  surface  of  the  earth  to  that  of  land 
lie  as  lOi  to  5,  and  the  mean  depth  of  the  sea  be  a  quarter  of  a  mile  ;  how 
many  years  would  it  tnke  if  the  ocean  were  empty  to  fill  it  by  the  rivers 
running  at  the  present  rate  ?         Ans.  1708  years,  17  days,  aiid  12  hours. 

54.  If  a  cubic  foot  of  water  neigh  1000  o^.  avoirdupois,  and  the  weight 
of  mercury  be  13,V  times  greater  than  of  water,  and  the  height  of  the  mer- 
cury in  the  barometer  (tiie  weight  of  which  ts  equal  to  the  weight  of  acol- 
nmn  of  air  on  the  same  base,  extending  to  the  top  of  the  atmosphere)  be  30 
inches  ;  what  will  be  the  weight  of  the  air  upon  a  squarq  foot  ?  a  square 
mile  ?  and  what  will  be  the  whole  weight  of  the  atmosphere,  suppoeingthe 
size  X)f  the  earth  as  in  questions  51  and  53? 

,  Ans.    _     2109,375  poujiJs  weight  on  a  square  foot. 

52734375000         ...       square  miles, 
1 0240980468750000000     -         -         -     of  the  Zihole  atmosphert. 


£08  MISCELLANEOUS  QUESTIONS. 

65.  A  br^n  trido  June  ],  nith  40  dolhrs,  and  took  in  F.  3^  a  partner, 
Pept.  8,  followins:,  viili  120  dollar?  ;  on  Dec.  24,  A  put  in  190  dollars  more, 
and  continued  tfic  ivhole  in  traile  till  ATny  5,  following,  when  their  whole 
(Tsin  was  found  to  he  2^2  dblbrs ;  what  is  each  partner's  share  ? 

.^7).'.  Ji's  aharc  ;^47,066+/^'-^  "'•"'re  ^34,9.>44- 

56.  Tf  I  give  HO  bushels  of  potatoes  at  21  cents  per  bushel,  and  240/6.  of 
flax,  at  15  cents  per  lb.  for  G4  bushels  of  salt,  what  is  the  salt  per  hnshel  ? 

.'Ins.   1^0,025. 

57  Whpt  is  the  present  ivorth  of  4C2  dollars,  payable  4  years  hence,  dis- 
counting' at  the  rate  of  6  per  cent  ?  Arts.  ^088,709. 

53.  i  have  owina:  to  me  ns  follows  :  vi/.  ^18,73  in  8  months  :  $46,00  in 
."i  raontli?  ;  and  104,84  in  P>  months  ;  what  is  the  mean  lime  for  the  payment 
of  the  whole  ?  .Ins.  4  inonths  2  days. 

59.  If  I  sell  500  {\oA^  at  \'jd.  a  piece,  and  lo^c  £,2  per  cent,  what  do  I 
lose  in  the  whole  quantity  I  Ans.  £,2  IBs.  3d. 

no.  If  I  buy  1000  El!=i  Tlemi'!)  of  iirr^n  for  £.90,  what  may  I  sell  it  pfr 
ni  in  London,  to  piin  £10  in  the  whf'o  '!  Ans.  3s.  4f/. 

61.  Ilovv  many  wine  p;alIons  in  a  cask,  whose  bunpf  diameter  measures 
C7  iuchct.  head  diamotcr  2]  inches,  and  length  30  inches? 

^.  .ins.  63  gals.  3g1s. 

r>?.  A  ni'iitnrv  viTiri^y  drovv  !ip  hi--,  uoldif^rs  in  rank  and  tile,  having  the 
P'unb*  r  in  r.iuk  and  file  erjir.d  ;  on  beinj^  reititbrced  with  tliree  times  hi? 
first  number  of  men,  he  placed  them  all  in  the  same  form,  and  then  the 
number  in  rank  and  tile  was  just  double  what  it  was  at  first ;  he  was  again 
reinforced  with  three  times  his  first  number  of  men,  and  after  placing  the 
whole  in  the  same  form  as  at  first,  his  number  in  rank  and  file  was  40  men 
each  ;  How  miny  men  had  he  at  first?  Anx.  100. 

C3.  Two  ships  A  and  B  sailed  from  a  certain  port  at  the  same  time  ;  A 
sailed  north  8  miles  an  hour,  and  B  east  6  miles  an  hour  ;  What  was  their 
distance  at  the  end  of  one  h6ur?  Avs.  10  miles. 

64.  A  hare  starts  12  rods  before  a  bound  ;  but  is  not  perceived  by  him 
>3ntil  she  has  been  up  45  seconds  ;  she  scuds  away  at  the  rate  of  ten  mile*; 
an  hour,  and  the  dog  on  view,  makes  after  at  the  rate  of  16  miles  an  hour. 
How  lung  will  the  hound  be  in  overtaking  the  hare,  and  what  distance  will 
hfi  run  ?  Ans.  97^  seconds,  he  Zinll  rvn  22iiGfeef. 

Go.  A  fellow  said  that  when  he  counted  his  nuts  two  by  two,  three  by 
tliree,  four  by  four,  five  b}'  five,  and  six  by  six,  there  was  still  an  odd  one  ; 
b-it  when  he  counted  them  seven  by  seven  they  came  out  even  ;  How  ma- 
ny had  he  ?  *  Ans.  721. 

66.  There  is  an  island  r)0  miles  in  circumference,  and  three  men  start 
toi^ether  to  travel  the  same  way  about  it ;  A  goes  7  miles  per  daj',  B  8,  and 
C  [> ;  when  will  they  all  come  together  again,  and  how  flir  will  they  trav- 
el ?         Ans.  £>0  days.  A  350  tniles.     B  400,  and  C  "450. 

TT.  If  a  weight  of  14401b.  be  placed  1  foot  from  the  prop,  at  what  dis- 
tance fiom  the  prop  must  a  power  of  I60lb.  be  applied  to  balance  it? 

Ans.  9  fret. 
G3.  Sound,  ui^ir)tcrnTpled,  moves  about   1142  feet  in  a  second  ;.suppos- 
ir.'^  in  a  thundor  storm,  ihe  space  between  the  hghtning  and  thunder  be  six 
seconds  :  at  what  distance  was  the  explosion  ? 

Ahs.  I  mxlc,  04roffs,  2 1 /«/'/. 


^ 


MISCELLANEOUS  QUESTIONS.  209 

CD.  A  cannon  ball  at  the  first  discharge,  flies  about  a  mile  in  eight  seconds; 
at  this  rate,  how  long  would  a  ball  be  in  passing  from  the  Earth  to  the  Sun, 
it  being,  as  astronomers  well  know,  95 173000  miles  distant  ? 

Ans.  24  Tjears,  46  days,  7  hours,  33m.  iOs. 

70.  A  general  disposing  his  army  into  a  square  battalion,  found  he  had 
531  over  and  above  ;  but  increasing  each  side  with  one  soldier,  he  wanted 
■4i  to  fill  up  tb«  square  ;  Of  how  many  men  did  his  army  consist  ? 

Ans.  19000. 

71.  A  and  B  cleared  by  an  adventure  at  sea  45  guineas,  which  was  £35 
per  cent  upon  the  money  advanced,  and  with  which  they  agreed  to  purchase 
a  genteel  horse  and  carriage,  whereof  they  were  to  have  the  use  in  pro- 
portion to  the  sums  adventured,  which  was  found  to  be  11  to  A,  as  often  as 
<j  to  B  ;  what  money  did  each  adventure  ? 

Ans.  A  £104  4$.  2\^d.     B  £75  15*.  9j\d. 

72.  Tubes  may  be  made  of  gold  weighing  not  more  than  at  the  rate  of 
■YSiH  °^  *  grain  per  foot ;  what  would  be  the  weight  of  such  a  tube,  which 
would  extend  across  the  Atlantic  from  Boston  to  London,  estimating  the 
distance  at  3000  miles  I 

Ans.  2oz.  6pwt.  S/y^^r. 


PLEASIJVG  AND  DIVERTING  qUESTIONS. 


1.  There  was  a  well  30  feet  deep,;  a  frog  at  the  bottom  could  jump  up 
3  feet  every  day,  but  he  would  fall  back  two  feet  every  night.  How  many 
days  did  it  take  the  frog  to  jump  out  ? 

2.  Two  men  were  driving  sheep  to  market,  says  one  to  the  other,  give  me 
one  of  yours  and  I  shall  have  as  many  as  you  ^  the  other  says,  give  me  one 
of  yours  and  I  shall  have  as  many  again  as  you.     How  many  had  each  ? 

3.  As  I  was  going  to  St.  Ives, 
I  met  seven  wives. 
Every  wife  had  seven  sacks, 
Every  sack  had  seven  cats, 
<Every  cat  had  seven  kits, 
Kits,  cats,  sacks  and  wives, 
How  many  were  going  to  St.  Ives  ? 

4.  The  account  of  a  certain  school  is  as  follows,  viz.  1-16  of  the  boys 
learn  geometry,  3-8  learn  grammar,  3-10  learn  arithmetic,  3-20  to  write, 
and  9  learn  to  read  ;  I  demand  the  number  of  each  ? 

5.  A  man  driving  his  geese  to  market,  was  met  by  another,  who  said, 
Good  morrow  master,  with  your  hundred  geese  ;  says  he,  I  have  not  an 
hundred,  but  if  I  had  half  as  many  as  I  nov/  have,  and  two  geefe  and  a 
half  beside  the  number  I  nor/  have  a!roady,  1  should  have  an  hundred. 
How  many  bad  he  .' 

C  c  • 


210  MISCELLANEOUS  QUESTIONS. 

6.  Three  travellers  met  at  a  Caravansary,  or  inn  in  Persia  ;  and  two  of 
them  brought  their  provision  along  with  them,  according  to  the  ctistom  of 
the  country  ;  but  the  third  not  having  provided  any,  proposed  to  the  others 
tliat  they  should  eat  togetl>er,  and  he  would  pay  the  value  of  his  proportion. 
This  being  agreed  to,  A  produces  5  loaves,  and  B  3  loaves,  which  the 
travellers  eat  together,  and  C  paid  3  pieces  of  money  as  the  value  of  his 
share,  with  which  the  others  were  satisfied,  but  quarrelled  about  the  di- 
viding ('fit.  U})on  this,  the  affair  was  referred  to  the  judge,  who  decided 
the'  dispute  bj'  an  impartial  sentence.     Reqtiired  his  decision  ?  ^f^'^  / 

7.  Suppose  the  9  digits  to  be  placed  in  a  quadrangular  form ;  I  demand 
in  wliat  order  they  must  stand,  that  any  three  figures  in  a  right  line  may 
make  just  16  ? 

l'.  a  countryman  having  a  Fox,  a  Goose,  and  a  peck  of  corn,  in  his 
journey,  came  to  a  river,  where  it  so  happened  that  he  could  carry  bat  one 
over  ai  a  time.  Now  as  no  two  were  to  be  left  together  that  might  de- 
slro}  each  other  ;  so  lie  was  at  his  wits  end  how  to  dispose  of  them  ;  for 
s;iys  iie,  tho'  tlie  corn  can't  eat  the  goose,  nor  the  goose  eat  the  fox ;  yet 
tiitj  fox  can  eat  the  goa-se,  and  the  goose  eat  the  com.  The  question  is, 
how  he  must  carry  them  over  that  they  may  not  devour  each  other  ? 

9.  Three  jealous  husbands  with  their  wives,  being  ready  to  pass  by 
night  over  a  river,  do  find  at  the  water  side  a  boat  which  can  carry  but  two 
persons  at  once,  and  for  want  of  a  waterman  they  are  necessitated  to  row 
tiienjselves  over  the  river  at  several  times  :  The  question  is,  how  those 
six  persons  shall  pass  by  2  and  2,  so  that  none  of  the  three  wives  may  be 
found  in  the  company  of  one  or  two  men,  unless  her  husband  be  present  ? 

10.  Two  merry  companions  are  to  have  equal  shares  of  8  gallons  of 
wine,  which  are  in  a  vessel  containing  exactly  8  gallons;  now  to  divide  it 
eqsially  between  them,  they  have  only  two  other  empty  vessels,  of  which 
one  contains  5. gallons,  and  the  other  3  :  The  question  is,  how  they  shall 
divide  the  said  wine  botweon  them  by  the  help  of  these  three  vessels,  so 
that  tboy  may  liave  four  gallons  apiece  ? 


SECTION  IV. 

.^j,  FQRilS  OF  NOTES,  DEEDS,  BONDS,  AND  OTHER  INSTRUMENTS  OF 
J,  *  WRITING. 


§  1.  OF  NOTES. 

Ab.  /. 

Overdean,  Sept.  17,  1802. — For  value  received  I  promise  to  pay  to  OH- 
ter  Bountiful,  or  order,  sixty  three  dollars'fifty  four  cents,  ott demand,  with 
interest  after  three  months.  William  Trusty. 

Attest,  Timothy  Testimony. 

No.  11. 

^    Bilfort,  Sept.  17,  1802. — For  value  received,  I  promise  to  pay  to  O.  R. 

or  bearer • — dollars cents,  three  months  after  date. 

Peter  Pencil. 

By  two  Persons, 
.irian,  Sept.   17,  1802. — For  value  received,  we  jointly  and  severally 

promise  to  pay  to  C.  D.  or  order, dollars, cents,  on  demand 

with  interest. 

Attest,  Constance  Mley.  Alden  Faithful. 

James  Faikfacb. 

OBSERV.^TIONS. 

1 .  No  note  is  negotiable  unless  the  words,  or  order,  other\*T«e  or  bearer j 
be  inserted  in  it. 

2.  If  the  note  be  written  to  pay  him  ^' or  order,^^  (No.  1.)  then  Oliver 
Bountiful  may  endorse  this  note,  that  is,  write  his  name  on  the  backside 
and  sell  it  to  A,  B,  C,  or  whom  he  pleases.  Then  A,  who  buys  the  note, 
calls  on  William  Trusty  for  payment,  and  if  he  neglects,  or  is  unable  to 
pay,  A  may  recover  it  of  the  endorser. 

3.  If  a  note  be  written  to  pay  him  "  of  bearer,^*  (No.  2.)  then  any  per- 
son who  holds  the  note  may  sue  and  recover  the  same  of  Peter  Pencil. 

4.  The  rate  of  interest  establislied  by  law,  being  s/x  ^5er  cent  per  annum, 
it  becomes  unnecessary  in  w  riting  notes  to  mention  the  rate  of  interest  •,  it 
»s  sufficient  to  write  them  for  the  payment  of  such  a  sum,  with  interest,  for 
it  will  ^e  understood  legal  interest,  which  is  six  per  cent. 

.5.  All  notes  ai-e  either  payable  on  demand  or  at  the  expiration  of  a  cer- 
tain term  of  time  agreed  upon  by  t)ie  parties  and  mentioned  in  the  note,  as 
three  months,  a  year,  &:c. 

6.  If  a  bond  or  note  mentioU  no  time  of  payment,  it  is  always  on  de- 
mand, whether  the  words,  on  dnncnd,  be  expressed  of  not. 


21S  F0R3IS  or  BONDS. 

7.  AH  notes  pa3'abic  at  a  certain  time  ar«  or  iaterest  as  soon  a.'  they  be- 
come due,  thoug;!!  in  ench  notes  there  be  no  mention  made  ol' interest. 

This  rule  is  founded  on  the  pr;ncij)lo  that  every  man  ought  to  receire 
Lis  money  when  due,  and  that  the  noo  payment  oth  at  that  time  is  an  inju- 
ry to  him.  The  law,  therefore,  to  do  him  justice,  allows  him  interest  irom 
the  time  the  money  becomes  due,  as  a  compensation  for  the  injury. 

y.  Upon  the  same  principle  a  note  payable  en  demand,  without  any  men- 
tion made  of  intere^st,  is  on  interest  aAcr  a  demand  of  payment,  for  upoa 
demand  such  notes  immediately  become  due. 

9.  If  a  note  be  given  for  a  speciric  article,  as  rye,  payable  in  one,  two, 
or  three  months,  or  in  any  certain  time,  and  the  signer  of  such  note  suffers 
the  time  to  elapse  without  delivering  such  ai-licle,  the  holder  of  the  note 
will  not  be  obliged  to  take  the  article  afterwards,  but  may  demand  and  re- 
cover the  value  of  it  in  money. 

^   2.  OF  BONDS. 

A  BOND  WITH  A  CONDITION  FROM  ONE  TO  ANOTHER. 

KNOW  all  men  by  these  presents,  that  I,  C.  D.  of  &c.  in  the  county  of  &c. 
atn  hel.l  and  iirndy  bound  to  E.  F.  of  &c.  in  two  hundred  dollars,  to  be  paid 
to  the  said  E.  F.  or  his  certain  attorney,  his  executors,  admiuistratoi-s  or 
jwsigns,  to  which  payment,  well  and  truly  to  be  made,  I  bind  myself,  my 
heirs  executors  and  administrators,  firmly  by  these  presents ;  Sealed  with 
u.y  seal.  Dated  tlve  eleventh  day  of in  the  year  of  our  Lord  one  thou- 
sand eigiit  hundred  ar.d  two. 

The  Condilion  of  this  obligation  is  such,  that  if  the  above  bound  C.  D. 
his  heirs,  executor?,  or  adaiinistrators,  do  and  shall  well  and  truly  pay,  or 
cause  to  be  paid  unto  the  above  named  E.  F.  his  executors,  administrators 
cr  assigns,  the  fLjll  sum  of  two  hundred  dollars,  with  legal  interest  for  the 

same,  on  or  before  the  eleventh  day  of next  ensuing  the  date  hereof  : 

Tlien  this  obligation  to  be  void,  or  otherwise  to  remain  in  full  force  and 
virtue. 

Signed,  4"C. 

A  Condition  of  a  Cowiter  Bond,  or  Bond  of  Indemnity ,  where 
one  man  becomes  bound  for  another. 

THE  condition  of  this  obligation  is  such,  that  whereas  the  above  named 
A.  B.  at  the  special  instance  and  request,  and  for  the  only  proper  debt  of 
{lie  above  bound  C.  D.  together  with  the  said  C.  D.  is,  and  by  one  bond  or 
obli.ofation  bearing  equal  date  with  the  obligation  above  v/ritten,  held  and 

iirmly  bound  unto  E.  F.  of  &.c.  in  the  penal  sum  of dollars, 

conditioned  for  the  payment  of  the  sum  of,  &c.  with  legal  interest  for  the 
same,  on  the day  of next  ensuing  the  date  of  the  said  in  part  reci- 
ted obligation,  as  in  and  by  the  said  in  part  recited  bond,  with  the  condition 
thereunder  written  may  more  fully  appear:  If  therefore  the  said  C.  D.  his 
heirs,  executors,  or  administrators,  do  and  shall  well  and  truly  pay,  cr  cause 
to  be  paid  unto  the  said  E.  F.  his  executors,  administratox-s,  or  assigns,  the 

said  sum  of,  iic.  with  legal  interest  of  the  same,  on  the  said day  of,  &lc. 

next  ensuing  the  date  of  the  ?aid  ia  part  recited  obligation,  according  to  the 
l.rue  intent  i'nd  meaning,  and  in  full  discharge  and  satisfaction  of  the  said  in 
part  re.:iiou  bond  or  oblt;;ution  :    Then,  o:c,  Otherwise,  &ic. 


FORMS  OF  RECEIPTS.  2 IS 

Note.  The  principal  difference  between  a  note  and  a  bond,  is  that  the 
latter  is  an  instrument  of  more  solemnity,  being  given  under  seal.  Also,  a 
note  may  be  controuled  bj'  a  special  agreement,  different  from  the  note, 
whereas  in  case  of  a  bond,  no  special -agreement  can  in  the  least  controul 
wliat  appears  to  have  been  the  intention  of  liie  parties  as  expressed  by  the 
words  in  the  condition  oi'  the  bond. 

i^  3.  OF  RECEIPTS. 
No,  I. 

Sitgrieves,  Sept.  IP,  1802.  Received  from  Mr.  Durance  Adleg^  ten 
dollars  in  full  of  all  accounts. 

Orvand  Constance. 

No.  II. 

Sitgrieves,  Sept.  19,  1802.  Received  of  Mr  Orvand  Constance,  five 
dollars  in  full  of  all  accounts. 

Durance  Adlev. 

No.  III. 

Receipt  for  an  cndorscnicjit  on  a  Note. 

Sitgrieves,  Sept.  19,  1802.  Received  of  Mr.  Simpson  Easily,  (by  the 
hand  of  Titus  Trusty)  sixteen  dollars  twenty  five  cents,  which  is  enilorscd 
on  his  note  of  June  3,  1802. 

Peter  Cheerfi'l. 

No.  IF, 

A  Receipt  for  money  received  on  account, 

Sttgrieves,  Sept.  19,  1802.  Received  of  Mr.  Orand  Landike,  fifty  dol- 
lars on  account. 

Eldro  Slacklev. 

No,  V. 

Receipt  for  interest  due  on  a  Bond, 

Received  this -day  of of  Mr.  A.  B.  the  sum  of  five  pounds  in 

full  of  one  year's  interest  of  £100  due  to  me  on  the day  of last 

on  bond  from  the  said  A.  B.     I  say  received.         By  roe  C.  D. 

OBSERrATIONS. 

1.  There  is  a  distinction  between  receipts  given  in  full  of  nil  accounts, 
and  others  in  full  of  all  demands.  The  former  cut  oil  accounts  nnly ;  the 
latter  nit  off  not  onlv  all  account*,  but  all  obligations  an:l  ri-?;bt  ot  action. 

2.  When  any  two"  persons  make  a  settlement  and  pass  receipts  (No.  I. 
and  11.)  each  receipt  must  9]^ec\fy  a  particular  ?nm  recpived,  less  or  more 
it  is  not  iiecessarv  that  the  sum  specified  in  tlic  receipt,  be  the  exact  sum 
'received. 


£11  FORMS  OF  ORDERS. 

^  4.  OF  ORDERS. 
No.  I. 

Mr.  Stephen  Burgess, 
Sir, 
For  value  received,  pay  to  A.  B.  Ten  Dollars,  and  place  the  same  to 
my  acco'ut.  Samuel  Skinner. 

Archdale,  Sept.   P,    1802. 

No.  IL 

Sir,  BostotiySept.  9,  1802v 

For  value  received,  pay  G.  R.  eighty  six  cents,  and  this  with  his  receipt 
shall  be  your  discharge  from  me. 

Nicholas  Reubens. 
To  Mr.  James  Robottom, 

^  5.  OF  DEEDS. 
No.  L 

A   Warrantee  Deed. 

K.voAv  ALL  MEN  BY  THFst:  PRESENTS,  That  I,  Peter  Careful,  of  Leomin- 
ster, in  the  County  of  Worcester,  and  Commonwealth  of  Massachusetts, 
gentlecian,  for  and  in  consideration  of  one  hundred  and  fifty  dollars,  and 
forty  five  cents,  paid  to  me  by  Samuel  Pendleton  of  Ashby,  in  the  County 
of  Midf'lesex,  and  Commonwealth  of  Massachusett.?,  yeoman,  the  receipt 
w!;  reof  I  do  hereby  acknowledge,  do  hereby  give,  grant,  sell  and  convey 
to  i!ie  said  Pamnel  Pendleton,  his  heirs  and  assigns,  a  certain  tract  and  par- 
cel of  hnd,  bounded  as  follows,  viz. 

[Here  insert  the  bounds,  together  with  all  the  privileges  and  appurtenanceg 
thereunto  ^efoa^ing.] 

To  have  and  to  hold  the  same  unto  the  said  Samuel  Pendleton,  his  heirs 
and  assit^ps  to  his  and  their  use  and  behoof  forever.  And  I  do  covenant 
with  the  said  Samuel  Pendleton,  his  heirs  and  assigns,  that  I  am  lawfully 
seis;ed  in  see  of  the  premises,  that  they  are  free  of  all  incumbrances,  and 
tliat  I  will  warrant  and  will  defend  the  same  to  the  said  Samuel  Pendleton. 
liis  heirs  and  assigns  forever,  against  the  lawful  claims  and  demands  of  all 
persons. 

In  witness  whereof,  I  hereunto  set  my  hand  and  seal  this day  of — 

in  the  year  of  our  Lord  one  thousand  eight  hundred  and  two. 

Signed,  sealed  and  delivered  i  Peter  Careful,     O 

in  presence  of  ) 

L.  R.        F.  G. 

No.  IL 

Quitclaim  Deed. 

Kn'ow  all  men  by  these  presents.  That  I,  A.  C.  of,  &c.  in  considera- 
tion of  the  sum  of to  be  paid  by  C.  D.  of&c.  the  receipt  whereof  I  do 

hereby  acknowledge,  have  remissed,  released,  and  forever  quitclaimed,  and 
do  by  these  presents  remit,  release,  and  forever  quitclaim  unto  the  said 
C.  D.  his  heirs  and  assigns  forever  {Here  insert  the  premises.')  To  have 
and  to  hold  the  same,  together  with  all  the  privileges  and  appurtenanceg 
thereunto  belonging,  to  him  the  said  C.  D.  his  heirs  and  assigns  forever. 
— In  n'iinessy  ^c. 


FORMS  OF  DEEDS.  ?16 

No.  III.  A  Mortgage  Deed. 

Know  all  men  by  these  presents,    That  I  Simpson  Easley,    of 

in  the  County  of in  the  State  of Blacksmith,  in  consideration 

of Dollars Cents,  paid  by  Elvin  Fairface  of in  the 

county  of in  the  State  of Shoemaker,  the  receipt  whereof  L  do 

hereby  acknowledge,  do  hereby  give,  grant,  sell  and  convey  unto  the  said 
Elvin  Fairface,  his  heirs  and  assigns,  a  certain  tract  and  parcel  of  land, 
bounded  as  follows,  viz.  (^Here  insert  the  bounds,  together  uiih  all  the 
privileges  and  appurtenances  thereunto  belonging.')  To  have  and  to  hold 
the  afore  granted  premises  to  the  said  Elvin  Fairface,  his  heirs  and  assigns, 
to  his  and  their  use  and  behoof  foiever.  And  1  do  covenant  with  the  said 
Elvin  Fairface,  his  heirs  and  assigns,  That  I  am  lawfully  seized  in  fee  of 
the  afore  granted  premises.  That  they  are  free  of  all  incumbrances : 
That  I  have  good  right  to  sell  and  convey  the  same  to  the  said  Elvin  Fair- 
face.  And  that  I  will  warrant  and  defend  the  same  premises  to  the  said 
Elvin,  his  heirs  and  assigns  forever,  against  the  lawful  claims  and  demands 
of  all  persons.  Provided  nevertheless,  That  if  I  the  said  Simpson  Easley, 
iny  heirs,  Executors,  or  administrators  shall  well  and  truly  pay  to  the 
eaid    Elvin  Fairface,   his  heirs,  executors,  administrators  or  assig^ns,  the 

full  and  just  sum  of dollars cents  on  or  before  the day 

of which  will  be  in  the  year  of  our  Lord  eighteen  hundred  and 

with  lawful  interest  for  the  same  until  paid,  then  this  deed,  as  also  a  cer- 
tain bond  [or  note,  as  the  case  may  be]  bearing  even  date  v\ith  these  pre- 
sents given  by  me  to  the  said  Fairface,  conditioned  to  pay  the  same  sum 
and  interest  at  the  time  aforesaid,  shall  be  void,  otherwise  to  remain  in  full 
force  and  virtue.  In  witness  whereof,  I  the  said  Simpson  and  Abigail  my 
wife,  in  testimony  that  she  relinquishes  all  her  right  to  dower  or  alimon}' 
in  and  to  the  above  described  premises,  hereunto  set  our  hands  and  seals 

this day  of in  the  year  of  our  Lord  one  thousand  eight  hundred 

and  five. 

'  Signed,  sealed  and  delivered  }  Simpson  Easley.  O 

in  presence  of  y  Abigail  Easley.  O 

L.  N.    V.  X. 

^  6.  OF  AN  INDENTURE. 
A  common  Indenture  to  hind  an  Apprentice. 

THIS  Indenture  witnesseth,  That  A.  B.  of,  kc.  hath  put  and  placed,  and 
by  these  presents  doth  put  and  bind  out  his  son  C.  D.  and  the  said  C.  D. 
doth  hereby  put,  place  and  bind  out  himself,  as  an  apprentice  to  K.  P.  to 

learn  the  art,  trade,  or  mystery  of The  sal  1  C.  D.  after  the  manner 

of  an  apprentice,  to  dwell  with  and  serve  the  said  R.  P. from  the  day 

of  the  date  hereof,  until  the day  of which  will  be  in  the  vt.ar  c^ 

our  Lord  one  thousand  eight  hundred  and at  which  time  the  Sixi't  ap- 
prentice, if  he  should  be  living,  will  be  twenty  one  years  of  age  :  During 
which  time  or  term  the  said  apprentice  his  s-.nd  master  well  and  frtithfu'ly 
shall  serve  ;  his  secrets  keep,  and  his  lawful  commands  every  where,  r.nd 
at  all  times  readily  obey.  He  shall  do  no  damage  to  his  said  mastiff,  nor 
wilfully  suffer  any  to  be  done  by  others  ;  and  if  any  to  his  kncnUdg^:^  be 
intended,  he  shall  give  his  master  seasonable  novice  thereof.  lie  sImI!  not 
waste  the  goods  of  his  said  master  nor  lend  them  unlavvfiiHy  lo  any  ,  at 
cards,  dice,  or  any  unlawful  gan.  >,  he  shall  not  play  ;  fornication  he  sh<)li 
not  commit,  nor  matrimony  contract  during  the  said  term;  taverns,  ale 
houses,  or  places  of  gaming  he  shall  not  hauat  or  fiequent :  From  the  ser- 


216  FORM  OF  A  WILL. 

vice  of  his  said  master  lie  fhall  not  absent  hi!r.?f!lf ;  but  in  all  tiling;,  nnd  al 
all  times  hs  shall  cariv  and  beliave  himself  as  a  good  and  faithful  aj'prcDtice 
ought,  duri'.ig  the  whole  lime  or  term  aforesaid. 

And  the  said  K.  P.  on  his  part  dot'.i  hereby  promise,  covenant  and  ai^ree 
to  teach  and  instruct  the  said  appienti':e,  or  cause  him  to  be  taught  and  in- 
structed in  the  art,  trade  or  calliu^^  of  a— ———'by  the  bojt  \ray  or  nicars 
he  can,  and  also  teach  and  instruct  th'^  said  apprentice,  or  cause  him  to  be 
tanc^ht  and  instructed  to  read  and  wiile,  and  CA'puer  ds  fir  as  the  rule  of 
Three,  if  the  said  apprentice  be  capable  to  learn,  pnd  shall  nell  and  failh- 
full}'  find  and  proviJe  for  the  said  apprentice,  p^ood  and  sufficient  merit, 
drink,  cloathiiig,  lodging  and  other  necessaries  tit  and  convenient  for  sucli 
an  apprentice,  during  the  term  aforesaid,  and  at  the  exp'iration  thereof, 
shall  give  unto  t!v3  said  apprentice,  two  suits  of  wcaiing  apparel,  one  suit- 
able for  the  Lord's  day,  and  the  other  for  working  days. 

In  testimony   whereof',  the  said  )',artie8  have  liereunto  interchangeably" 

set  their  hands  and  seals,  this  said day  of in  the  year  of  our 

Lord  one  tho'isand  eight  hundred  and (Seal) 

Sipied,  scaled  and  delivered  }  (Seal) 

in  presence  of  y  (Seal) 

^  7.  OF  A  WILL. 
The  form  of  a  Will  mth  a  Devise,  of  a  Heal  Estate,  Lease- 

koldy  Si'C. 
I.i  the  r.ame  of  God,  Jlmfn,  I,  A.  B.  of,  &c.  being  weak  in  body,  but  of 
sound  and  perfect  mind,  and  memor\',  {or  yov  may  say  ifms,  considering  the 
tjncertairity  of  this  mortal  life,  and  being  of  sound,  Lc.)  blessed  be  Almightjf 
God  for  the  same,  do  make  and  publish  this  as  my  last  Will  and  Testament  in 
a  manner  and  form  following  (that  is  to  s&y)  Firsts  I  give  and  bequeath  un- 
to my  beloved  wife,  J.  B.  the  sum  of 1  do  also  give  and  bequeath  un- 
to my  eldest  son  G.  B.  the  sum  of 1  do  also  give  and  bequeath  un- 
to my  two  younger  sons  J.  B.  and  F.  B.  the  sum  of^ apiece.     1  also 

give  and  beqneatii  to  my  daughter  in  law,  S.  H.  H.  single  woman,  the  sum 

of which  said  several  legacies  or  sums  of  money,  I   will  and  order 

shall  be  paid  to  the  said  respective  legatees  within  six  months  after  my  de- 
cease. I  further  give  and  devise  to  my  said  eldest  son  G.  B.  his  heirs  and 
assigns,  All  that  my  messuage  or  tenement,  situate,  lying  and  being  in  &c. 
together  with  all  my  other  freehold  estate  v,  halsoever,  to  hold  to  him  the 
said  G.  B.  his  heirs  and  af?signs  forever.  And  I  hereby  give  and  bequeath 
to  my  said  younger  song  J.  B.  and  F.  B.  all  my  leasehold  estate  of  and  in  all 
those  messuages  or  tenements,  with  the  appurtenances,  situate  &.c.  equally 
to  be  divided  between  them.  Ar.d  lastW,  as  to  all  the  rest,  residue  and  re- 
mainder of  my  pen-ional  estate,  goods  and  chattels,  of  what  kind  and  nature 
foever,  I  give  and  bequeath  the  same  to  my  ssid  beloved  wife  J.  B.  whom 
I  hereby  appoint  sole  executrix  of  thi'^  my  last  Will  and  Testament  ;  and 
hereby  revoking  all  former  Wills  by  me  made. 

In  zvitness  n-hcr€'cf,  1  hereimto  set  tny  hand  and  seal,  (his     -  ■  ■  day  of 

in  the  year  of  our  Lord 

Signed,  denied,  paliliphed  ftriii  tlec'.ared  by  llie  nbovn  nnmcri  A.  B.  (Seal) 

A.  B.  to  Le  Lis  last  A\ill  and  Tfstament  in  the  presf.'ico  of  ns, 
who  linvc  liereunto  subscrilied  our  napies  ?,■<  \\iti)e.=si>--,  in  the 
presence  of  lue  testator,  11.  S. 

V.-.  T. 

'J .  \\\ 


APPENDIX. 


VULGAR  FRACTIONS. 

ULGAR  FRACTIONS  are  parts  of  an  unit,  or  inte- 
fi^c: ;  and  arc  represented  by  two  numbers,  placed  one  above 
the  other,  with  a  line  drawn  between  them. 

The  number  above  the  line  is  called  the  numerator,  and  that  below  the 
line  the  denominator. 

The  denominator  shews  how  many  parts  an  integer  is  divided  into, 
and  the  numerator  shews  how  many  of  those  parts  are  meant  by  the  frac- 
tion. 

Fractions  are  either  proper,  improper,  compound,  or  mixed. 

A  proper  fraction  is  when  the  numerator  is  less  than  the  denomina- 
tor •  a'?  1  -5   11   &,c 

An  improper  fraction  is  when  the  numerator  is  greater  than  the  de- 
rominalor  ;  as  f ,  |,  IJ.^  ^c. 

A  compound  fraction  is  the  fraction  of  a  fraction,  coupled  by  the  word 
oj;  as  a  of  |,  ^c. 

A  mixed  number  or  fraction  is  composed  of  a  whole  number  and  a 
fraction  ;  as  7  f ,  28  |,  &c. 

Reduction  of  Vulgar  Fractions. 

J.  To  reduce  a  given  fraction  to  its  lowest  terms  ^ 

RULE. 

Divide  both  the  numerator  and  the  denominator  by  some  one  number 
ihat  will  divide  them  both  without  a  remainder  :  divide  the  quotients 
in  the  same  manner,  and  so  on  till  no  number  greater  than  1  will  divide 
ihem  both,  and  the  last  quotients  express  the  fraction  iu  its  lowest  terms. 

1.  Reduce  jif  to  its  lowest  terms. 

4)     3) 
Thus,  8  f  |i=3A=^j._^a  the  Answer. 

2.  Reduce  ^ff  to  its  lowest  terms.  ^ns.  |. 
.'^.  Reduce  ^\^  to  its  lowest  terms.  •5«s.  \. 

I.  Reduce  ^eif-  to  its  lowest  terms.  iJ>'W-  {h 

D  2  .    " 


218  VULGAR  FRACTIONS.  Appenbix 

Or  ; — Find  a  common  measure,  thus, 
Divide  the  denomiuator  by  the  numerator,  and  that  divisor  by  the  re- 
mainder,   continuing  so  to  do  till  nothing  remains  ;    the    last  diviscr  is 
tlie  common  nieasui-e  ;  then  divide  both  terms  of  the  fraction  b}  the  com- 
mon measure,  *  and  the  quotients  will  express  the  fraction  reqiiired. 

1.  Reduce  ||||'  to  its  lowest  terms. 
1080)1224(1 
1080 

144)1080(7 
1008 


Common  Measure.         72)144(2 
144 
Then  72)ifi|(|4  the  Ansziuer. 

2.  Reduce  ^|^^  to  its  lowest  terms.  Ans.  Jj'^. 

//.   To  reduce  a  mixed  number  to  an  improper  fraction. 

RULE. 
Multiply  the  whole  number  by  the  denominator  of  the  fraction,  and 
to  the  product  add  the  numerator  for  a  new  numerator,  and  place  it  over 
the  denominator. 

1.  Reduce  127y\  to  an  improper  fraction. 

J-  Here  we  multiply  the  whole  nurn- 

ber,   127  by  17  the  denominator  of  the 

ooQ  fraction,    adding   in  the  numerator  4  ; 

,aj  the  sum  2163  is  the  numerator  to  the 

xr.,«,«^„+«-  ..  fraction  soua;ht,  and  17  the  denomina- 

JNumerator,         4  .  xu  *  o,«t  •    *u    •  f 

tor,  so  that  21^3  ig  the  improper  irac- 

2  if  3  Ans.  ^'°°'  equal  to  127^^. 

2.  Reduce  653  ^-^  to  an  improper  fraction.  Ans.  '-/^». 

///.  To  reduce  an  improper  fraction  to  its  proper  terms,  or  mixed 

number. 
RULE. 
Divide  the  numerator  by  the  denominator,    the   quotient  will  be  the 
t\'hole  number,  and  the  remainder,  if  any,  will  be  the  numerator  to  the 
given  denominator. 

EXAMPLES. 

1.  Reduce  y  to  a  mixed  number. 

6)15(2|  ^?i<t. 

2.  Reduce  ^.m  to  its  proper  terms.  Ans.  27|. 

3.  Reduce  ^ubs  to  a  mixed  number.  Ans.  127^'^: 

4.  Reduce  ^Yir'"  to  a  mixed  number.  Ans.  653y=V' 

*  If  the  common  measure  happens  to  be  1.  the  fraction  is  already  in  its  lowest 
terms. 


Appendix  VULGAR  FRACTIONS.-,  219 

IF.  To  find  the  least  common  multiph  of  two  or  more  numbers. 

RULE. 

Divide  by  any  number  that  will  divide  two,  or  more,  of  the  given 
numbers  %vithout  a  remainder,  and  set  the  quotients,  together  with  the 
undivided  numbers,  in  a  line  underneath. 

Divide  these  quotients  and  undivided  numbers  as  before,  and  so  on, 
till  there  are  no  two  numbers  that  can  be  divided  ;  then  the  continued 
product  of  the  divisors  and  the  last  quotients  together  will  be  the  least 
common  multiple  required. 

EXAMPLES. 

1.  What  is  the  least  common  multiple  of  9,  8,  15  and  16? 

8)     9     8     15     16 
3)     9     i      15       2~ 


3     16       2 
Then  8  X3  X  3  X  5  X  2=720  .4ns. 

2.  What  is  the  least  number  that  3,  5,  8,  and  10  will  measure  ? 

Ans.  120. 

3.  What  is  the  least  number  that  can  be  divided  by  the  nine  digits 
^vithout  a  remainder  '  Ans.  2520. 

V.  To  reduce  a  compound  fraction  to  a  single  one. 

RULE. 

MnUiply  all  the  numerators  together  for  a  new  numerator,  and  all  the 
denominators  for  a  new  denominator,  then  reduce  the  new  fraction  to  its 
fowest  term  by  Case  I. 

EXAMPLES. 

1.  Reduce  ^  of  |  of  y'^  to  a  single  fraction. 

3x5x9 

=134=JL.  Anszeer. 

4X6X10" 

2.  Reduce  f  of  |  of  if  to  a  single  fraction.  .Ins.  |H. 

3.  Reduce  j\  of  i|  of  ^  of  20  *  to  a  simple  (or  improper)  fraction. 

finf     eS4n O  3_ 

*^'""    3  06  0  51* 

VI.  To  reduce  fractions  of  different  denominations  to  equivalent  fractions 
having  a  common  denominator. 

RILE. 

Multiply  each  numerator  into  all  the  denominators,  except  its  own,  for 
a  new  numerator,  and  aU  the  denominators  into  each  other,  continually, 
for  a  common  denominator. 


*  Any  whole  number  may  he  reduced  to  an  improper  fraction,  by  placing  1  un- 
der it  for  a  deaoniinator,  which  mu«st  be  done  in  this  case,  thus  3-". 


220  VULGAR  FRACTIONS.  Appendix. 

EXAMPLES. 

J.  Reduce  i,  |  and  |  to  equivalent  fractions,  having  a  common  denomi- 
nator. 

I  X  5  X  8=40  the  n€w  numerator  for  i 
2x4x8=C4  fori 

,r,x4xo=K>0  for  f 

4x^X8=160  the  common  denominator.  •  -' 

Hence  the  new  equivalent  fractions  are  y'-''^,  /j^"^  and  {^^  the  answer. 

2.  Reduce  a,  §  of  |,  7J,  and  J^  to  a  common  denominator. 

.IMS.  fa,  J,   187-i?    T3T2  '  TafS- 

3.  Reduce  -f^,  |  of  2^,  J-^  and  |  to  a  common  denominator. 

*'^''*-   TiiS.i'   TfilTT'  TlfSu^  ^""   "rfSSo- 

VII.  To  reduce  a  fraction  of  one  Senomination  to  the  fraction  of  miother,  hut 

greater,  retaining  the  saute  talue. 

RDLE. 

Reduce  the  given  fraction  to  a  compound  one  by  companng  it  with  all 
the  denominations  between  it  and  that  denomination  you  would  reduce  it 
to  ;  then  reduce  that  compound  fraction  to  a  simple  one,  by  Case  V. 

EXAMPLES. 

1.  Reduce  \  of  a  penny  to  the  fraction  of  a  pound. 

By  comparing  it,  the  compound  fraction  will  be  ^  of  Jj  of  jL.  Then 
"iX    IX    1 

=-J--  of  a  poii/nd,  the  Answer. 

8X12X20 

2.  Reduce  ^  of  a  pound  Avoirdupois  to  the  fraction  of  1  cwt. 

'  Ans.  ylg-  Ck'/. . 

3.  Reduce  |  of  a  Pennyweight  to  the  fraction  of  a  pound  Troy. 

Ans.  ji-  lb. 

4.  Reduce  |  of  a  penny  to  the  fraction  of  a  Guinea. 

Ans.  ^5^  firifiinea. 

VIII.  To  reduce  the  fraction  of  one  denomination  to  the  fraction  of  another, 

hut  less,  retaini/ig  the  same  value. 

JIULE. 
Reduce  the  given  fraction  to  a  compound  one,  as  in  the  preceding  case, 
only  observing  to  invert  the  parts  contained  in  the  integer  ;  then  reduce 
thi:  compound  fraction  to  a  simple  one,  and  that  to  its  lowest  terms. 

EXAMPLES. 

J.  Reduce  ^0^0  °^^  pound  to  the  fraction  of  a  penny.  ■ 

7     V  20  V  1 2  ^ 

Thus  ^./^  0  of  V  of  '/ .  Then  --—  -— — ==J5|  o==,i  d. 
T.20        1         1  1920X1X1      '^"'     * 

2.  Reduce  ^Jj-^  of  a  pound  to  the  fraction  of  a  firthing.  Ans.  ^q. 

.'3.  Reduce  frj-^n  of  a  ft  Troy  to  the  fraction  of  a  p'Ji't.  ._^      Ans.  I  pwt. 
4.   Reduce  -^  *  of  a  guinea  to  the  fraction  of  a  pound.  Ans.  a£. 

*  C'omjjared  thus  ?  of  ^y  of  ^'-. 


Appendix  V  OT^GaH  FRACTIONS.  tH 

IX.   To  find  the  value  nf  a  fraction  tn  the  known  parts  rf  the  mtcgier. 

RULE. 
Multiply  the  numerator  by  the  piirts  in  the  next  inferior  denomination, 
and  divide  the  product  by  the  denuminator  ;  and  it' any  thin^;  reiaiiin,  mul- 
tiply it  by  the  next  inferior  denomination  and  divide  as  before,  and  io  ou^ 
as  far  as  necessary  ;  the  several  quotients  will  be  the  answer. 

EXA3IPLES. 

r.  What  is  the  value  of  §  of  a  pound  ? 

2  "  Multiply  the  numerator  of   the 

20  fraction  ('J)  by  20,  the  number  of 

3\.jQ  shillingSiiu  a  pound  ;  the   product, 

— ~r  J  (40)  divided  by  3,  the  denominator, 

.(,  gives  13,  the   number  of  shillings, 

1  and  1  remaining;,  being  mullijdicd  by 

^)^^  12  the  number  of  pence  in  a  shiliins;, 

4d.  and  the  product,  (12)  divided  as  bc- 

Ans.  13s.  4d.  fore  by  3,  gives  4,  the  number  of 

pence,  and  no  remainder.      Hence 

the  auswer,  13s.  4t/. 

2.  What  is  the  value  of  5  of  a  pound  Troy  ?  .2ns.  7  oz.  4  pz^t. 

3.  What  is  tlie  value  of  *  of  a  pound  Avoirdupois  ? 

Ans.  12  oz.  12}  dr. 

4.  Reduce  l  of  a  mile  to  its  proper  quantity.         Ans.  6  fur.  16  poles. 

H)  ( "io  ! 
l'.  To  reduce  any  given  (jitantity  to  the  fraction  of  a  greater  denomination  of 

thi  same  kitui. 

RULE. 

Reduce  the  given  quantit^f^  to  the  lowest  denomination  mentioned  for  a 
numerator  ;  then  reduce  the  integral  part  to  the  same  denomination  for  a 
denominator,  which  placed  under  the  uumerator  before  found  will  express 
the  fraction  required. 

EXAm'LES. 

1.  Reduce  16s.  Sd.  to  the  fraction  of  a  pound. 

!£.  Integral  part.  16s.  Sd. 

20  /  12 


20  200  Xumerator. 

12 

240  Denominator.  Ans.  |i^=|£' 

£.  Reduce  G  furlongs  and  IG  poles  to  the  fraction  of  a  mile. 

.i«s.  \  of  a  mile. 

3.  Reduce  12  or.  12^  dr.  to  the  fraction  of  a  pound  Avoirdupois. 

Ans.  {  of  a  poimtZ 


22?  VULGAR  FRACTIONS.  Api'f.ndix. 

Addition  of  Vulgar  Fractions. 

RULE. 

Reduce  compound  fractions  to  single  ones,  mixed  numbers  to  improper 
fractions,  and  tractions  of  different  integers  to  those  of  the  same,  and  nU 
of  them  to  a  common  denominator  ;  then  the  sum  of  tlie  nameratdr9 
written  over  the  common  denominator  will  be  the  sum  of  the  fractioB 
required. 

EXAMPLES. 

1.  Add  ^,  9  i,  and  |  of  i  together. 

First,  the  mixed  number  9}='/  ;  the  compound  fraction  |  of  |==|. 
Then  the  fractions  are  ^,  y  and  |  ;  which  reduce  to  a  common  denomi- 
nator. 

3x5x6:=     90 
40x7x6=1932 
2  X  7  X  5=_^ 
7  X  5  X  6=VtV=5vH  ^insrcer. 

2.  Add  1  Jj,  2^«^,  3^^  and  llj^  toget-her.  Ans.  22. 

3.  Add  i£.  -fs.  and  fd.  together.  Ans.  2s.  Qf%*jd. 

4.  Add  f  of  17£.  9f£.  and  |  of  i  of  ^£.  together.    .  t mi  ai  cu  i>r:>-i...o'iu 

Ans.  16£.  12s.  34rf. 


Subtraction  of  Vulgar  Fractions 

RULE. 

Prepare  the  fractions  as  in  addition,  and  ti.e  difference  of  the  numera- 
tors written  above  the  common  denominator  will  give  the  difference  of  the 
fi'actions  required. 

To  subtract  a  fraction  froin  a  n;hole  number;  from  the  denominator  of 
tlie  fraction  subtract  the  numerator  and  place  the  remainder  over  the  de- 
nominator :  then  deduct  1  from  the  integer  or  whole  number. 

EXAMPLES. 

1.  From  If  take  ^. 

49  X  9=441 
5x50=250 

50  X  9=:450  com.  denom. 

Therefore  ||-i=|||=i|L  the  Answer. 
£.  From  13  J  take  |  of  15.  Ans.  2Jj. 

3.  From  '{£.  take  |  of  a  shilling.  Ans.  14*.  3a'. 

4  From  7  weeks  take  9^^  days.  Am.  5a'.  4J.  7fe.''12^! 

5  From  o  take  -,-}  .  Ans.  9.\-^. 


Appendix.  VULGAR  FRACTIONS.  223 

Multiplication  of  Vulgar  Fractions. 

RULE. 

Reduce  compound  fractions  to  simple  ones,  and  mixed  numbers  to  im- 
proper fractions  ;  then  multiply  the  numerators  together  for  a  new  nume- 
rator, and  the  denominators  for  a  new  denombator. 

EXAMPLES. 

1.  Multiply  4i  by  j. 

U=%.   Then£2ll=-i!L  the  Answer. 
2X8 

2.  Multiply  1  of  5  by  f  of  a.  -^ns.  y^. 

3.  Multiply  48 1  by  13f  Ans.  GTS-j'^. 

4.  Multiply  /^  by  I  of  f  off  Ans.  ^^ 


Division  of  Vulgar  Fractions. 

RULE. 

Prepare  the  fractions,  as  already  directed  ;  then  invert  the  divisor  aad 
proceed  as  in  multiplication. 

EXAMPLES. 

1.  Divide  A  by  f. 

The  divisor  ^  iuverted  will  be  f .  Theni^=4f=f  ^"J- 

.'  7X2 

2.  Divide  5  by  J_.  Ms.  7f 

3.  Divide  9^  by  i  of  7.  Ms.  2if 

4.  Divide  f  by  9.  Ms.  ^. 

5.  Divide  7  by  f  Ans.  ISf. 

6.  Divide  52051  by  f  of  91.  Ms.  7U. 

Rule  of  Three  Direct  in  Vulgar  Fractions 

RULE. 

Having  stated  the  question,  make  the  necessary  preparations,  as  in  Re- 
duction of  Fractions,  and  invert  the  first  term  ;  then  proceed  as  in  Multi- 
Jt  plication  of  Fractions. 

EXAMPLES 

1.  If  i  of  a  yard  cost  §  of  a  pound,  what  will  2  of  *^  ell  English  come  to, 
at  the  same  rate  ? 


224  VULGAR  FRACTiONi:.  Appendix. 

First,  reilace  the  J  of  a  yard  to  the  fraction  of  an  Ell  Eiigliih  ;  thus  i  ot' 

E.  E.         £.  E.  E. 

Thien,  as     ^^     :     §•     ::     |.     Invert  the  first  term,  and  pro. 
Geed  as  in  Multiplication — 

Thus!22ll2iP-=i22f=£o.     0     0. 
4X3X5=  60= 

2.  If -5  yd.  cost  l£.  what  will  40^  yds.  come  to  ?  Ans.  £59  1«.  3(''. 

3.  If -j^  of  a  ship  cost  £51,  what  are  f.r  of  her  worth  ? 

Ans.£\0  18s.  Gd.  3}q. 

4.  At  £3|  per  Cu-t.  what  will  9|  lb.  come  to  ?  dns.  Qs.  3^d. 

5.  Iff  yd.  cost  ^  of  a  £.  what  will  /j-  Ell  Eoglish  cost  ? 

Ans.  lis.  Id.  2f«3. 

C.  A  man  owning  |  of  a  farm,  sells  |  of  his  share  for  £171  ;  what  is  the 
whole  farm  valued  at  ?  Ans.  £380. 


Rule  of  Three  Inverse  in  Vulgar  Fractions. 
EXAMPLES. 

1.  If  25?s.  will  pay  for  the  carriage  of  an  Cwt.  1451  nniles,  how  far  ma} 

6i^Cwt.  be  carried  for  the  same  money  ?  Ans.  22Jy  miles. 

2.  If  the  penny  white-loaf  weigh  7  oz.  when  a  bushel  of  wheat  cost  5s.  6d. 
what  is  the  bushel  worth  when  the  penny  white-loaf  weighs  2i  oz.  f 

Ans.  15s.  \U{. 

3.  How  much  shalloon,  that  i3  j  yard  wide^  will  line  6^  yards  of  cloth  that 

is  Ji  yard  wide  ?  .3ns.  11|  yards. 


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